# Wachs permutations, Bruhat order and weak order

Francesco Brenti and Paolo Sentinelli Dipartimento di Matematica, Universitá
di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Roma, Italy.
<EMAIL_ADDRESS>Dipartimento di Matematica, Politecnico di Milano,
Milan, Italy<EMAIL_ADDRESS>
###### Abstract

We study the partial orders induced on Wachs and signed Wachs permutations by
the Bruhat and weak orders of the symmetric and hyperoctahedral groups. We
show that these orders are graded, determine their rank function, characterize
their ordering and covering relations, and compute their characteristic
polynomials, when partially ordered by Bruhat order, and determine their
structure explicitly when partially ordered by right weak order.

##  1 Introduction

Wachs permutations are a class of permutations first introduced (in the even
case) in [22] to study the signed Eulerian numbers and the signed major index
enumerator of the symmetric groups. This class was extended in [4] to the odd
case and to signed permutations in order to study the enumerators of the odd
and even major indices of classical Weyl groups twisted by their one-
dimensional characters. In this work we study Wachs (and signed Wachs)
permutations in their own right. More precisely, we study the partial orders
induced on them by the Bruhat and weak orders of the symmetric and
hyperoctahedral groups. These orders are fundamental objects in algebraic
combinatorics and have important connections to algebra and geometry (see,
e.g., [2, Chaps.2 and 3], [1, Chap.2], [16, Chap.1], [11, Chap.5], [7,
Chap.10], [17, Chap.2], and the references cited there). Many subsets of the
symmetric and hyperoctahedral groups have been studied as posets under the
Bruhat order such as, for example, quotients, descent classes, and generalized
quotients [3], complements of quotients [19], involutions [9, 12, 13, 15, 18],
conjugation-invariant sets of involutions [5, 8], and twisted identities [10].
In this paper we show that Wachs permutations possess many nice properties
when partially ordered by Bruhat and weak orders. More precisely, we show that
they form a graded poset, determine their rank function, characterize the
ordering and covering relations, and compute the characteristic polynomial,
when partially ordered by Bruhat order, and determine their structure
explicitly when partially ordered by right weak order.

The organization of the paper is as follows. In the next section we collect
some notation and results that are used in the sequel. In §3 we study the
poset obtained by partially ordering Wachs permutations with respect to Bruhat
order. We show that these posets are always graded (Theorem 3.9), characterize
their order and covering relations (Theorem 3.13 and Proposition 3.12), and
compute their characteristic polynomials (Corollary 3.16). In §4, using some
of the results in §3, we obtain analogous results for the poset induced on
signed Wachs permutations by Bruhat order. More precisely, we show that these
posets are graded (Theorem 4.10), characterize their order and covering
relations (Theorem 4.13, Proposition 4.2, Corollary 4.12, and Corollary 4.6),
and compute the characteristic polynomials (Corollary 4.16). In §5 we study
the posets obtained by partially ordering Wachs and signed Wachs permutations
under right weak order. We show that they are always isomorphic to the direct
product of a Boolean algebra with the weak order on the whole group in rank
$\left\lceil\frac{n}{2}\right\rceil$ (Theorems 5.1 and 5.3). Finally, in §6,
we discuss some conjectures and open problems arising from the present work
and the evidence that we have about them.

##  2 Preliminaries

In this section we recall some notation, definitions, and results that are
used in the sequel. As $\mathbb{N}$ we denote the set of non-negative integers
and as $\mathbb{P}$ the set of positive integers. If $n\in\mathbb{N}$,
$[n]:=\\{1,2,...,n\\}$; in particular $[0]=\varnothing$. For $n\in\mathbb{P}$,
in the polynomial ring $\mathbb{Z}[q]$ the $q$-analogue of $n$ is defined by
$[n]_{q}:=\sum\limits_{i=0}^{n-1}q^{i}$ and the $q$-factorial by
$[n]_{q}!:=\prod\limits_{i=1}^{n}[i]_{q}$. For a set $A$ and
$f:A\rightarrow\mathbb{N}$ we define

$A(x,f):=\sum\limits_{w\in A}x^{f(w)}$

and $f(u,v):=f(v)-f(u)$, for all $u,v\in A$. The cardinality of a set $X$ will
be denoted by $|X|$ and the power set of $X$ by $\mathcal{P}(X)$. Given a
cartesian product $X\times Y$ of two sets $X$ and $Y$, we indicate with
$\pi_{1}$ and $\pi_{2}$ the projections on $X$ and $Y$ respectively.

Let $n\in\mathbb{N}$, $i\in\mathbb{Z}$, $q\in\mathbb{Q}$ and $J\subseteq[n]$;
then we define $J_{e}:=\\{j\in J:j\equiv 0\mod 2\\}$, $J_{o}:=\\{j\in
J:j\equiv 1\mod 2\\}$, $J+i:=\\{i+j:j\in J\\}\cap[n]$, and
$qJ:=\bigcup\limits_{j\in J}\\{qj\\}\subseteq\mathbb{Q}$.

We follow Chapter 3 of [20] for notation and terminology concerning posets. We
just recall some definitions. Given two posets $P$ and $Q$, their _ordinal
product_ $P\otimes Q$ is defined by ordering the set $P\times Q$ in the
following manner: $(x,y)\leqslant(x^{\prime},y^{\prime})$ if and only if
$x=x^{\prime}$ and $y\leqslant y^{\prime}$ or $x<x^{\prime}$, for all
$(x,y),(x^{\prime},y^{\prime})\in P\times Q$. If $\mu$ is the Möbius function
of a graded poset $P$ with minimum $\hat{0}$, maximum $\hat{1}$ and rank
function $\rho$, then the characteristic polynomial of $P$ in the
indeterminate $x$ is defined by

$C_{P}(x):=\sum\limits_{z\in P}\mu(\hat{0},z)x^{\rho(z,\hat{1})}.$

Next recall some basic results in the theory of Coxeter groups which will be
useful in the sequel. The reader can consult [2] or [11] for further details.
Let $(W,S)$ be a Coxeter system. The length of an element $z\in W$ with
respect of the given presentation is denoted as $\ell(z)$. If $J\subseteq S$,
we let

$\displaystyle W^{J}$ $\displaystyle:=$ $\displaystyle\\{w\in
W:\ell(ws)>\ell(w)~{}\forall~{}s\in J\\},$ $\displaystyle{{}^{J}W}$
$\displaystyle:=$ $\displaystyle\\{w\in W:\ell(sw)>\ell(w)~{}\forall~{}s\in
J\\},$ $\displaystyle D(w)$ $\displaystyle:=$ $\displaystyle\\{s\in
S:\ell(ws)<\ell(w)\\},$

and, more generally, for any $A\subseteq W$ we let $A^{J}:=A\cap W^{J}$. The
subgroup $W_{J}\subseteq W$ is the group generated by $J$. In particular
$W_{S}=W$ and $W_{\varnothing}=\\{e\\}$, being $e$ the identity of $W$. We
consider on $W$ the Bruhat order $\leqslant$ (see, e.g., [2, Chapter 2] or
[11, Chapter 5]) and on any subset we consider the induced order. When the
group $W$ is finite, there exists a unique maximal element $w_{0}$ of maximal
length. We recall this characterizing property of the Bruhat order, known as
the _lifting property_ (see [2, Proposition 2.2.7]):

######  Proposition 2.1.

Let $v,w\in W$ be such that $v<w$ and $s\in D(w)\setminus D(v)$. Then
$v\leqslant ws$ and $vs\leqslant w$.

For $J\subseteq S$, each element $w\in W$ has a unique expression
$w=w_{J}w^{J}$, where $w^{J}\in W^{J}$ and $w_{J}\in W_{J}$ (see [2,
Proposition 2.4.4]). Often we consider the projection $P^{J}:W\rightarrow
W^{J}$ defined by $P^{J}(w)=w^{J}$. This map is order preserving ([2,
Proposition 2.5.1]). We let $T:=\\{wsw^{-1}:w\in W,s\in S\\}$. The following
lemma will be useful in the next section; for a proof see [2, Lemma 2.2.10].

######  Lemma 2.2.

Suppose that $x<xt$ and $y<ty$, for $x,y\in W$, $t\in T$. Then, $xy<xty$.

For $w\in W$ we set $T_{L}(w):=\\{t\in T:\ell(tw)<\ell(w)\\}$; the _right weak
order_ $\leqslant_{R}$ on $W$ is the partial order whose cover relations are
defined by letting $u\vartriangleleft_{R}v$ if and only if $v^{-1}u\in S$ and
$\ell(v)=\ell(u)+1$, for all $u,v\in W$ (see [2, Chapter 3]). Then the
following characterization holds (see [2, Proposition 3.1.3]):

######  Proposition 2.3.

Let $(W,S)$ be a Coxeter system. Then $u\leqslant_{R}v$ if and only if
$T_{L}(u)\subseteq T_{L}(v)$, for all $u,v\in W$.

Now, for any $n\in\mathbb{P}$, let $S_{n}$ be the group of all bijections of
the set $[n]$. It is well known that it is a Coxeter group with set of
generators $\\{s_{1},s_{2},...,s_{n-1}\\}$, being $s_{i}$ the simple inversion
given, in one line notation, by $12...(i+1)i...n$. Given a permutation
$\sigma=\sigma(1)\sigma(2)...\sigma(n)\in S_{n}$, the action of $s_{i}$ on the
right is given by $\sigma
s_{i}=\sigma(1)\sigma(2)...\sigma(i+1)\sigma(i)...\sigma(n)$, for all
$i\in[n-1]$. For $i,j\in[n]$, the action on the right of a transposition
$(i,j)$, $i\neq j$, then is given by
$\sigma(i,j)=\sigma(1)\sigma(2)...\sigma(i-1)\sigma(j)...\sigma(j-1)\sigma(i)...\sigma(n)$,
for all $\sigma\in S_{n}$. As a Coxeter group, identifying
$\\{s_{1},s_{2},...,s_{n-1}\\}$ with $[n-1]$, we have that (see e.g. [2,
Propositions 1.5.3 and 1.5.2])
$D(\sigma)=\\{i\in[n-1]:\sigma(i)>\sigma(i+1)\\}$ and
$\ell(\sigma)=\ell_{A}(\sigma)$, where
$\ell_{A}(\sigma):=\operatorname{inv}(\sigma)=|\\{(i,j)\in[n]^{2}:i<j,\sigma(i)>\sigma(j)\\}|$,
for all $\sigma\in S_{n}$. Then, given $J\subseteq[n-1]$,
$S^{J}_{n}=\\{\sigma\in S_{n}:\sigma(i)<\sigma(i+1)~{}\forall~{}i\in J\\}$.

For $i\in[n]$ and $A\subseteq S_{n}$ define $A(i):=\\{\sigma\in
A:\operatorname{pos}(\sigma)=i\\}$, being
$\operatorname{pos}:S_{n}\rightarrow[n]$ the function defined by
$\operatorname{pos}(\sigma):=\sigma^{-1}(n)$, for all $\sigma\in S_{n}$. We
find it convenient to define the following involution on $[n]$:

$i^{*}:=\left\\{\begin{array}[]{ll}i-1,&\hbox{if $i\equiv 0\mod 2$;}\\\
i+1,&\hbox{if $i\equiv 1\mod 2$ and $i+1\in[n]$;}\\\
n,&\hbox{otherwise,}\end{array}\right.$

and the simple inversion $s^{*}_{i}:=(i,i^{*})$, for all $i\in[n]$. Given a
permutation $\sigma\in S_{n}$, $k\in[n]$ and $i\in[k]$, define $\sigma_{i,k}$
as the $i$-th element in the increasing rearrangement of
$\\{\sigma(1),\sigma(2),...,\sigma(k)\\}_{<}$ and $\sigma^{-1}_{i,k}$ as the
position of $\sigma(i)$ in $\\{\sigma(1),\sigma(2),...,\sigma(k)\\}_{<}$. So
$\sigma_{j,k}=\sigma(i)$ if $j=\sigma^{-1}_{i,k}$. The following theorem,
known as the “tableau criterion”, characterizes the Bruhat order on $S_{n}$
(see [2, Theorem 2.6.3]).

######  Theorem 2.4.

Let $n\in\mathbb{P}$ and $\sigma,\tau\in S_{n}$. The following are equivalent:

  1. 1.

$\sigma\leqslant\tau$;

  2. 2.

$\sigma_{i,k}\leqslant\tau_{i,k}$ for all $k\in D(\sigma)$ and $i\in[k]$;

  3. 3.

$\sigma_{i,k}\leqslant\tau_{i,k}$ for all $k\in[n-1]\setminus D(\tau)$ and
$i\in[k]$.

We indicate by $w_{n}$ the maximum of the poset $(S_{n},\leqslant)$, i.e., in
one line notation, $w_{n}=n...321$.

The _descent number_ and the _major index_ are the functions
$\operatorname{des}:S_{n}\rightarrow\mathbb{N}$ and
$\operatorname{maj}:S_{n}\rightarrow\mathbb{N}$ defined respectively by
$\operatorname{des}(\sigma):=|D(\sigma)|$ and
$\operatorname{maj}(\sigma):=\sum_{i\in D(\sigma)}i$. A famous result of
McMahon asserts that $\ell_{A}$ and $\operatorname{maj}$ are equidistribuited
over $S_{n}$ (see, e.g. [20, Proposition 1.4.6]), i.e.
$S_{n}(x,\ell_{A})=S_{n}(x,\operatorname{maj})$, and this rank-generating
function is known to be (see, e.g [20, Corollary 1.3.10])

$S_{n}(x,\ell_{A})=[n]_{x}!.$ (1)

Following [4] we define functions from $S_{n}$ to $\mathbb{N}$ by letting

$\operatorname{odes}(\sigma):=|D(\sigma)_{o}|,\;\;\;\operatorname{emaj}(\sigma):=\sum_{i\in
D(\sigma)_{e}}\frac{i}{2},$

for all $\sigma\in S_{n}$, and call these functions _odd descent number_ and
_even major index_ respectively.

Following [4], for $n\in\mathbb{P}$, we let

$\operatorname{\mathcal{W}}(S_{n}):=\\{\sigma\in
S_{n}:|\sigma^{-1}(i)-\sigma^{-1}(i^{\ast})|\leqslant
1\;\mbox{if}\;i\in[n-1]\\}$

and call the elements of $\operatorname{\mathcal{W}}(S_{n})$ Wachs
permutations. It is not hard to see that, if $n$ is even,
$\operatorname{\mathcal{W}}(S_{n})=\\{\sigma\in
S_{n}:|\sigma(i)-\sigma(i^{\ast})|\leqslant 1\;\mbox{if}\;i\in[n-1]\\}$.

Let $[\pm n]:=\\{-n,\ldots,-1,1,\ldots,n\\}$. We denote by $B_{n}$ the group
of bijective functions $\sigma:[\pm n]\rightarrow[\pm n]$ satisfying
$-\sigma(i)=\sigma(-i)$, for all $i\in[n]$. We use the window notation. So,
for example, the element $[-2,1]\in B_{2}$ represents the function
$\sigma:[\pm 2]\rightarrow[\pm 2]$ such that $\sigma(1)=-2=-\sigma(-1)$ and
$\sigma(2)=1=-\sigma(-2)$. We let
$\operatorname{\mathrm{Neg}}(\sigma):=\\{i\in[n]:\sigma(i)<0\\}$,
$\operatorname{neg}(\sigma)=|\operatorname{\mathrm{Neg}}(\sigma)|$,
$\operatorname{nsp}(\sigma):=|\\{(i,j)\in[n]^{2}:\;i<j,\;\sigma(i)+\sigma(j)<0\\}|$,
$s^{B}_{j}:=(j,j+1)(-j,-j-1)$ for $j=1,...,n-1$, $s_{0}:=(1,-1)$, and
$S_{B}:=\\{s_{0},s^{B}_{1},...,s^{B}_{n-1}\\}$. It is well known that
$(B_{n},S_{B})$ is a Coxeter system and that, identifying $S_{B}$ with
$[0,n-1]$, the following holds (see, e.g., [2, §8.1]).

######  Proposition 2.5.

Let $\sigma\in B_{n}$. Then
$\ell_{B}(\sigma)=\ell_{A}(\sigma)+\operatorname{neg}(\sigma)+\operatorname{nsp}(\sigma)$,
and $D(\sigma)=\\{i\in[0,n-1]:\sigma(i)>\sigma(i+1)\\}$, where $\sigma(0):=0$.

We denote by $\leqslant$ the Bruhat order of $B_{n}$ and by
$\sigma\mapsto\tilde{\sigma}$ the embedding $B_{n}\hookrightarrow S_{\pm n}$
(where $S_{\pm n}$ is the set of all bijections of $[\pm n]$) . The following
result is [2, Corollary 8.1.9].

######  Proposition 2.6.

We have that $\sigma\leqslant\tau$ in $B_{n}$ if and only if
$\tilde{\sigma}\leqslant\tilde{\tau}$ in $S_{\pm n}$.

The next result, which appears in [14, Theorem 5.5], characterizes the cover
relations in the Bruhat order of $B_{n}$. For $\sigma\in B_{n}$ and
$i,j\in[\pm n]$, $i<j$, we say that $(i,j)$ is a rise for $\sigma$ if
$\sigma(i)<\sigma(j)$ (i.e., if $(i,j)$ is not an inversion of $\sigma$).
Given a rise $(i,j)$ for $\sigma$ we then say, following [14], that $(i,j)$ is
central if $(0,0)\in[i,j]\times[\sigma(i),\sigma(j)]$, that $(i,j)$ is free if
there is no $i<k<j$ such that $\sigma(i)<\sigma(k)<\sigma(j)$, and that it is
symmetric if $i=-j$.

######  Theorem 2.7.

Let $n\in\mathbb{P}$ and $\sigma,\tau\in B_{n}$. Then $\sigma\lhd\tau$ in
Bruhat order if and only if either

  1. 1.

$\tau=\sigma(i,j)(-i,-j)$ where $(i,j)$ is a non-central free rise of $\sigma$
or

  2. 2.

$\tau=\sigma(i,j)$ where $(i,j)$ is a central symmetric free rise of $\sigma$.

Following [4], for $n\in\mathbb{P}$, we let

$\operatorname{\mathcal{W}}(B_{n}):=\\{\sigma\in
B_{n}:|\sigma^{-1}(i)-\sigma^{-1}(i^{\ast})|\leqslant
1\;\mbox{if}\;i\in[n-1]\\}$

and call the elements of $\operatorname{\mathcal{W}}(B_{n})$ signed Wachs
permutations.

Our aim in this work is to study the sets of Wachs and signed Wachs
permutations under the Bruhat order and the weak order.

##  3 Wachs permutations and Bruhat order

Let $(W,S)$ be a Coxeter system. For any independent set $I\subseteq S$ (i.e.
$st=ts$ for all $s,t\in I$) we define a subgroup of $W$ by

$G_{I}:=\\{w\in W:I^{w}=I\\},$

where $I^{w}:=\\{wsw^{-1}:s\in I\\}$. Note that $G_{I}$ is a subgroup of $W$,
and that, since $I$ is independent, $W_{I}\subseteq G_{I}$. Furthermore,
$W_{I}$ is normal in $G_{I}$.

######  Proposition 3.1.

$P^{I}(G_{I})$ is a subgroup of $G_{I}$, isomorphic to the quotient
$G_{I}/W_{I}$. In particular we have the group isomorphisms

$G_{I}\simeq P^{I}(G_{I})\ltimes W_{I}\simeq S_{2}\wr_{I}P^{I}(G_{I}).$

###### Proof.

Let $w\in G_{I}$ and write $w=w^{I}w_{I}$ where $w^{I}\in W^{I}$ and $w_{I}\in
W_{I}$. Let $s\in I$. Then, by our hypothesis,
$wsw^{-1}=w^{I}w_{I}s(w_{I})^{-1}(w^{I})^{-1}=w^{I}s(w^{I})^{-1}$ so $w^{I}\in
G_{I}$. So $P^{I}(G_{I})\subseteq G_{I}$. Let $u,v\in G_{I}$ and write
$u=u^{I}u_{I}$ and $v=v^{I}v_{I}$ where $u^{I},v^{I}\in W^{I}$ and
$u_{I},v_{I}\in W_{I}$, so $u^{I},v^{I}\in P^{I}(G_{I})$. Let $s\in I$. Then
there is $t\in I$ such that $v^{I}s=tv^{I}$. Furthermore, since
$u^{I},v^{I}\in W^{I}$, $\ell(tv^{I})>\ell(v^{I})$, and
$\ell(u^{I}t)>\ell(u^{I})$. Therefore, by Lemma 2.2 we obtain that
$u^{I}v^{I}s=u^{I}tv^{I}>u^{I}v^{I}$. Hence $u^{I}v^{I}\in W^{I}$. But, since
$v^{I}\in G_{I}$, there exists $\tilde{u}_{I}\in W_{I}$ such that
$uv=u^{I}v^{I}\tilde{u}_{I}v_{I}$ so $(uv)^{I}=u^{I}v^{I}$. Hence
$u^{I}v^{I}\in P^{I}(G_{I})$. In particular, $(u^{I})^{-1}=(u^{-1})^{I}$, so
$P^{I}(G_{I})$ is a subgroup of $G_{I}$, and $P^{I}:G_{I}\rightarrow
P^{I}(G_{I})$ is a surjective homomorphism whose kernel is $W_{I}$. The last
statements follow by the definitions. ∎

Let $m\in\mathbb{P}$, $W:=S_{2m}$. For any set $X$, we consider
$\mathcal{P}(X)$ as an abelian group, the operation being the symmetric
difference $+$, i.e. $A+B:=\left(A\setminus B\right)\cup\left(B\setminus
A\right)$, for all $A,B\in X$. Then it is straightforward to see that if we
take $I=\\{s_{i}:i\equiv 1\mod 2\\}\subseteq S$ then
$\operatorname{\mathcal{W}}(S_{2m})=G_{I}$. The group isomorphisms
$P^{I}(\operatorname{\mathcal{W}}(S_{2m}))\simeq S_{m}$ and
$W_{I}\simeq\mathcal{P}([m])$ then imply

$\operatorname{\mathcal{W}}(S_{2m})\simeq
S_{m}\ltimes\mathcal{P}([m])=S_{2}\wr S_{m}.$

Therefore $\operatorname{\mathcal{W}}(S_{2m})$ is isomorphic to the
hyperoctahedral group.

We consider $S_{n}$ with the Bruhat order and the subset
$\operatorname{\mathcal{W}}(S_{n})\subseteq S_{n}$ with the induced order. It
is not difficult to see, using Theorem 2.4 that
$P^{I}(\operatorname{\mathcal{W}}(S_{2m}))\simeq S_{m}$ as posets for all
$m>0$, where the set $S_{m}$ is ordered by the Bruhat order. Let $m>0$. For
$u\in S_{m}$ and $T\subseteq[m]$ let $\phi_{2m}^{-1}(u,T):=v$, where
$v\in\operatorname{\mathcal{W}}(S_{2m})$ is defined by

$v(2i-1)=\left\\{\begin{array}[]{ll}2u(i)-1,&\hbox{if $i\not\in T$,}\\\
2u(i),&\hbox{if $i\in T$,}\end{array}\right.$

and

$v(2i)=\left\\{\begin{array}[]{ll}2u(i),&\hbox{if $i\not\in T$,}\\\
2u(i)-1,&\hbox{if $i\in T$,}\end{array}\right.$

for all $i\in[m]$. The following result follows easily from our definitions
and Theorem 2.4, and its proof is omitted.

######  Proposition 3.2.

Let $m\in\mathbb{P}$. Then

  1. 1.

$\phi_{2m}$ is a bijection;

  2. 2.

$\phi_{2m}:\operatorname{\mathcal{W}}(S_{2m})\rightarrow
S_{m}\otimes\mathcal{P}([m])$ is order preserving;

  3. 3.

$\phi_{2m}^{-1}:S_{m}\times\mathcal{P}([m])\rightarrow\operatorname{\mathcal{W}}(S_{2m})$
is order preserving.

Moreover $\ell(v)=4\ell(\tau)+|T|$ if $\phi(v)=(\tau,T)$.

Figure 3 shows the Hasse diagram of
$(\operatorname{\mathcal{W}}(S_{5}),\leqslant)$. By the following example we
see that $\phi_{2m}$ and $\phi_{2m}^{-1}$ are not poset isomorphisms in
general.

${4321}$${3421}$${4312}$${3412}$${2143}$${1243}$${2134}$${1234}$ Figure 1:
Hasse diagram of $(\operatorname{\mathcal{W}}(S_{4}),\leqslant)$.

######  Example 3.3.

Let $m=3$. Then $(123,\\{1,2,3\\})\leqslant(132,\varnothing)$ in
$S_{m}\otimes\mathcal{P}([m])$ but
$\phi_{2m}^{-1}(123,\\{1,2,3\\})=214365\nleqslant
125634=\phi_{2m}^{-1}(132,\varnothing)$ in
$\operatorname{\mathcal{W}}(S_{2m})$.

Let $m=2$, $u=2143$ and $v=3412$. Then
$u,v\in\operatorname{\mathcal{W}}(S_{4})$, $u<v$ and
$\phi_{2m}(u)=(12,\\{1,2\\})\nleqslant(21,\varnothing)=\phi_{2m}(v)$ in
$S_{m}\times\mathcal{P}([m])$.

We consider now $\operatorname{\mathcal{W}}(S_{2m+1})$. For any $m>0$ it is
not difficult to see that the set $\operatorname{\mathcal{W}}(S_{2m+1})$ is
not a group. The previous general construction in Coxeter systems gives, for
$W=S_{2m+1}$, the group $G_{I}\simeq\operatorname{\mathcal{W}}(S_{2m})$ which,
as a set, can be included in $\operatorname{\mathcal{W}}(S_{2m+1})$. For
$n>1$, define a function
$\chi_{n}:\operatorname{\mathcal{W}}(S_{n})\rightarrow\operatorname{\mathcal{W}}(S_{n-1})$
by

$\chi_{n}(v)(i)=\left\\{\begin{array}[]{ll}v(i),&\hbox{if
$i<\operatorname{pos}(v)$;}\\\ v(i+1),&\hbox{if
$i\geqslant\operatorname{pos}(v)$,}\end{array}\right.$

for all $i\in[n-1]$, $v\in\operatorname{\mathcal{W}}(S_{n})$. Note that the
function $\chi_{n}$ is not necessarily order preserving. For example, if $m=2$
and $u=21345$, $v=51234$ then $u\leqslant v$ but $\chi_{n}(u)=2134\nleqslant
1234=\chi_{n}(v)$.

Let $\phi_{2m+1}:\operatorname{\mathcal{W}}(S_{2m+1})\rightarrow[m+1]\times
S_{m}\times\mathcal{P}([m])$ be the function defined by

$\phi_{2m+1}(v):=((\operatorname{pos}(v)+1)/2,\tau,T),$

if $\chi_{2m+1}(v)=(\tau,T)$, for all
$v\in\operatorname{\mathcal{W}}(S_{2m+1})$. For example,
$\phi_{2m+1}(4312756)=(3,213,\\{1\\})$. As in the even case, noting that
$u\leqslant v$ implies $\operatorname{pos}(v)\leqslant\operatorname{pos}(u)$,
we have the following result whose proof follows easily from our definitions
and Proposition 3.2.

######  Proposition 3.4.

Let $m>0$. Then

  1. 1.

$\phi_{2m+1}$ is a bijection;

  2. 2.

$\phi_{2m+1}:\operatorname{\mathcal{W}}(S_{2m+1})\rightarrow[m+1]^{*}\otimes
S_{m}\otimes\mathcal{P}([m])$ is order preserving;

  3. 3.

$\phi_{2m+1}^{-1}:[m+1]^{*}\times
S_{m}\times\mathcal{P}([m])\rightarrow\operatorname{\mathcal{W}}(S_{2m+1})$ is
order preserving,

where $i\leqslant j$ in $[m+1]^{*}$ if and only if $j\leqslant i$, for all
$i,j\in[m+1]$. Moreover $\ell(v)=4\ell(\tau)+|T|+2(m-i+1)$ if
$\phi_{2m+1}(v)=(i,\tau,T)$.

${54321}$${53421}$${54312}$${43521}$${53412}$${52143}$${34521}$${43512}$${51243}$${52134}$${34512}$${43215}$${21543}$${51234}$${34215}$${43125}$${12543}$${21534}$${34125}$${12534}$${21435}$${12435}$${21345}$${12345}$
Figure 2: Hasse diagram of $(\operatorname{\mathcal{W}}(S_{5}),\leqslant)$.

Let $n>0$ and $f_{n}:\operatorname{\mathcal{W}}(S_{n})\rightarrow
S_{\left\lfloor\frac{n}{2}\right\rfloor}$ be the function defined by the
assignment $(i,\tau,T)\mapsto\tau$ if $n$ is odd, and $(\tau,T)\mapsto\tau$ if
$n$ is even. By Proposition 3.2, if $n$ is even then $f_{n}$ is order
preserving. Corollary 3.8 states that the function $f_{n}$ is order preserving
also in the odd case. We also define a function
$\ell_{\operatorname{\mathcal{W}}}:\operatorname{\mathcal{W}}(S_{n})\rightarrow\mathbb{N}$
by

$\ell_{\operatorname{\mathcal{W}}}(v):=\ell(v)-\ell(f_{n}(v)),$

for all $v\in\operatorname{\mathcal{W}}(S_{n})$. For example,
$\ell_{\operatorname{\mathcal{W}}}(342156)=5-1=4$ and
$\ell_{\operatorname{\mathcal{W}}}(3472156)=9-1=8$. The minimum and the
maximum of the poset $(\operatorname{\mathcal{W}}(S_{n}),\leqslant)$ are
respectively the identity of $S_{n}$, which corresponds to $(e,\varnothing)$
in the even case and to $(m+1,e,\varnothing)$ in the odd one, and the element
of maximal length $w_{n}$ of $S_{n}$, which corresponds to $(w_{m},[m])$ in
the even case and to $(1,w_{m},[m])$ in the odd one, where
$m:=\left\lfloor\frac{n}{2}\right\rfloor$. Moreover
$\ell_{\operatorname{\mathcal{W}}}(e)=0$ and

$\ell_{\operatorname{\mathcal{W}}}(w_{n})={{n}\choose{2}}-{{\lfloor\frac{n}{2}\rfloor}\choose{2}}.$
(2)

Note that if $n$ is even then
$\ell_{\operatorname{\mathcal{W}}}(w_{n})=\frac{(3n-2)n}{8}=\frac{(3m-1)m}{2}$.
So $\\{\ell_{\operatorname{\mathcal{W}}}(w_{2m})\\}_{m\in\mathbb{P}}$ is the
sequence of pentagonal numbers (see A000326 in OEIS). If $n$ is odd then
$\ell_{\operatorname{\mathcal{W}}}(w_{n})=\frac{3(n^{2}-1)}{8}=\frac{3m(m+1)}{2}$;
so $\\{\ell_{\operatorname{\mathcal{W}}}(w_{2m+1})\\}_{m\in\mathbb{P}}$ is the
sequence of triangular matchstick numbers (see A045943 in OEIS).

Let $n>0$ and $m:=\left\lfloor\frac{n}{2}\right\rfloor$; for any $i,j\in[m]$,
$i<j$, we define an involution
$w^{A}_{i,j}:\operatorname{\mathcal{W}}(S_{n})\rightarrow\operatorname{\mathcal{W}}(S_{n})$
by

$w^{A}_{i,j}(v)=\left\\{\begin{array}[]{ll}(\tau(i,j),T+\\{i,j\\}),&\hbox{if
$n$ is even and $v=(\tau,T)$;}\\\ (k,\tau(i,j),T+\\{i,j\\}),&\hbox{if $n$ is
odd and $v=(k,\tau,T)$,}\end{array}\right.$

for all $v\in\operatorname{\mathcal{W}}(S_{n})$. For example, if
$v=4312765\in\operatorname{\mathcal{W}}(S_{7})$, then
$w^{A}_{2,3}(v)=4356721$.

######  Lemma 3.5.

Let $m>0$, $v=(\tau,T)\in\operatorname{\mathcal{W}}(S_{2m})$, $i,j\not\in T$
and $\tau(i,j)\vartriangleleft\tau$. Then $w^{A}_{i,j}(v)<v$ and
$\ell_{\operatorname{\mathcal{W}}}(v)-\ell_{\operatorname{\mathcal{W}}}(w^{A}_{i,j}(v))=1$.
If $u=(\sigma,S)\in\operatorname{\mathcal{W}}(S_{2m})$, $u<v$ and
$\sigma\leqslant\tau(i,j)$ then $u\leqslant w^{A}_{i,j}(v)$.

###### Proof.

We have that $\hat{v}:=w^{A}_{i,j}(v)=(\tau(i,j),T\cup\\{i,j\\})$. By the
tableau criterion it is easy to deduce that $\hat{v}<v$ and the equality
$\ell_{\operatorname{\mathcal{W}}}(\hat{v},v)=1$ follows easily from
Proposition 3.2 and the definition of $\ell_{\operatorname{\mathcal{W}}}$. We
now show that $u\leqslant\hat{v}$. We use Theorem 2.4. Note first that, since
$u$ and $\hat{v}$ are Wachs permutations and $\sigma\leqslant\tau(i,j)$,
$u_{l,h}\leqslant\hat{v}_{l,h}$ for all $1\leqslant l\leqslant h$ if $h$ is
even. Furthermore, since $u<v$, $u_{l,h}\leqslant\hat{v}_{l,h}$ for all
$l\in[h]$ if $h\leqslant 2i-2$ or $h\geqslant 2j$. So let $k\in[m]$ be such
that $k\not\in T$ and $i<k<j$. We wish to show that
$u_{l,2k-1}\leqslant\hat{v}_{l,2k-1},$ for all $1\leqslant l\leqslant 2k-1$.

Let $a:=v(2i-1)$ and $b:=v(2j-1)$. So $a+1=v(2i)$ and $a>b+1=v(2j)$ as well as
$\hat{v}(2i-1)=b+1$, $\hat{v}(2i)=b$, $\hat{v}(2j-1)=a+1$ and $\hat{v}(2j)=a$.
Let $c:=v(2k-1)$ and $d:=u(2k-1)$. Note that, since
$\tau(i,j)\vartriangleleft\tau$, $c\not\in[b,a+1]$ and that $c\equiv
1\operatorname{(\mathrm{mod}~{}2)}$ since $k\not\in T$. Let $r,s\in[k]$ be
such that $v_{2r-1,2k-1}=c$ and $u_{2s-1,2k-1}=d$. We have two cases to
consider according as to whether $c<b$ or $c>a+1$.

Say $c>a+1$.

Let $p,q\in[k]$ be such that $v_{2q-1,2k-1}=a$ and $\hat{v}_{2p-1,2k-1}=b$, so
$p\leqslant q$, $v_{2q,2k-1}=a+1$ and $\hat{v}_{2p,2k-1}=b+1$. Then $2r-1>2q$
since $c>a+1$. Note that

$v_{l,2k-1}=\hat{v}_{l,2k-1}$ (3)

if $1\leqslant l\leqslant 2p-2$ or $2q+1\leqslant l\leqslant 2k-1$ and that

$v_{l,2k-1}=\hat{v}_{l+2,2k-1}$ (4)

if $2p-1\leqslant l\leqslant 2q-2$. Moreover, since $u\leqslant v$ by our
hypothesis,

$u_{l,2k-1}\leqslant v_{l,2k-1},$ (5)

for all $1\leqslant l\leqslant 2k-1$.

Note that, by (3) and (5), we have

$u_{l,2k-1}\leqslant v_{l,2k-1}=\hat{v}_{l,2k-1}$

if $1\leqslant l\leqslant 2p-2$ or $2q+1\leqslant l\leqslant 2k-1$. So assume
$2p-1\leqslant l\leqslant 2q$.

If $x,y\in\mathbb{P}$, $x,y\equiv 1\operatorname{(\mathrm{mod}~{}2)}$, then we
find it convenient to write $\\{x,x+1\\}\leqslant\\{y,y+1\\}$ if $x\leqslant
y$ and similarly for $<$.

We have two cases to consider.

Let $s>q$. Then, since $\sigma\leqslant\tau(i,j)$, we have that

$\\{u_{2h-1,2k-1},u_{2h,2k-1}\\}\leqslant\\{\hat{v}_{2h-1,2k-1},\hat{v}_{2h,2k-1}\\}$

for all $p\leqslant h\leqslant q$, so $u_{l,2k-1}\leqslant\hat{v}_{l,2k-1}$
for all $2p-1\leqslant l\leqslant 2q$.

Let $s\leqslant q$. If $p\leqslant h\leqslant s-1$ then, since
$\sigma\leqslant\tau(i,j)$, we have that

$\\{u_{2h-1,2k-1},u_{2h,2k-1}\\}\leqslant\\{\hat{v}_{2h-1,2k-1},\hat{v}_{2h,2k-1}\\}$

so $u_{l,2k-1}\leqslant\hat{v}_{l,2k-1}$ for all $2p-1\leqslant l\leqslant
2s-2$.

We may therefore assume that $\max\\{2p-1,2s-1\\}\leqslant l\leqslant 2q$.
Since $\sigma\leqslant\tau(i,j)$, we have that

$u_{l,2k-2}\leqslant\hat{v}_{l,2k-2}$

for $1\leqslant l\leqslant 2k-2$. But

$u_{l,2k-1}=u_{l-1,2k-2}$

if $2s\leqslant l\leqslant 2k-1$ since
$\\{u(1),...,u(2k-2)\\}=\\{u(1),...,u(2k-1)\\}\setminus\\{d\\}$, and similarly

$\hat{v}_{l,2k-1}=\hat{v}_{l,2k-2}$

if $1\leqslant l\leqslant 2r-2$. Therefore

$u_{l,2k-1}<u_{l+1,2k-1}=u_{l,2k-2}\leqslant\hat{v}_{l,2k-2}=\hat{v}_{l,2k-1},$

if $2s-1\leqslant l\leqslant 2r-2$, so $u_{l,2k-1}\leqslant\hat{v}_{l,2k-1}$
if $2s-1\leqslant l\leqslant 2q$, because $2q\leqslant 2r-2$.

The case $c<b$ is similar (and slightly simpler) except that one uses $2k$
rather than $2k-2$ and obtains that

$u_{l,2k-1}=u_{l,2k}\leqslant\hat{v}_{l,2k}=\hat{v}_{l-1,2k-1}<\hat{v}_{l,2k-1}$

if $2p\leqslant l\leqslant 2s-1$. We leave the details to the interested
reader. ∎

The following lemma can be proved using Lemma 3.5 and the definitions.

######  Lemma 3.6.

Let $m>0$, $v=(k,\sigma,S)\in\operatorname{\mathcal{W}}(S_{2m+1})$,
$i,j,k\not\in S$ and $\sigma(i,j)\vartriangleleft\sigma$. Then
$w^{A}_{i,j}(v)<v$, $v(k,k+2)<v$ and
$\ell_{\operatorname{\mathcal{W}}}(w^{A}_{i,j}(v))=\ell_{\operatorname{\mathcal{W}}}(v(k,k+2))=\ell_{\operatorname{\mathcal{W}}}(v)-1$.

The next result is the crucial technical tool for the proof of our main
theorem, and states that elements of $\operatorname{\mathcal{W}}(S_{2m+1})$
are “join-irreducible” in a certain sense.

######  Proposition 3.7.

Let $m>0$, $u,v\in\operatorname{\mathcal{W}}(S_{2m+1})$, $u<v$ and
$i:=\operatorname{pos}(v)$. If $u<v$ and $\operatorname{pos}(u,v)>0$ then
$u\leqslant z$ where $z\in\operatorname{\mathcal{W}}(S_{2m+1})$ is defined by

$z=\left\\{\begin{array}[]{ll}v(i,i+2),&\hbox{if $v<v(i+1,i+2)$;}\\\
v(i+1,i+2),&\hbox{otherwise.}\end{array}\right.$

###### Proof.

Let $h:=(i+1)/2$ and consider first the case $i+1\not\in D(v)$. Let
$a:=v(i+1)$; then $a+1=v(i+2)=z(i)$, $a=z(i+1)$ and $z(i+2)=2m+1$. Define
$c:=u(i)$, $r\in[m+1]$ be such that $u(2r-1)=2m+1$, $p\in[h]$ be such that
$z_{2p-1,i}=a+1$ and $q\in[h]$ be such that $u_{2q-1,i}=c$. Notice that
$u(i+1)\in\\{u(i)+1,u(i)-1\\}$ since $\operatorname{pos}(u,v)>0$. Assume that
$q\leqslant p$. Note that

$u_{l,i}\leqslant v_{l,i}=z_{l,i}$

if $1\leqslant l\leqslant 2p-2$, while

$u_{l,i}=u_{l-1,i-1}\leqslant v_{l-1,i-1}=z_{l-1,i-1}=z_{l,i}$

if $2p\leqslant l$, and

$u_{2p-1,i}=\left\\{\begin{array}[]{ll}u_{2p-2,i-1}\leqslant
v_{2p-2,i-1}=z_{2p-2,i-1}=z_{2p-2,i}<z_{2p-1,i},&\hbox{if $q<p$,}\\\
u_{2p-1,i+1}\leqslant v_{2p-1,i+1}=a<a+1=z_{2p-1,i},&\hbox{if $q=p$ and
$u(i)<u(i+1)$,}\\\ u_{2p-1,i+1}+1\leqslant
v_{2p-1,i+1}+1=a+1=z_{2p-1,i},&\hbox{if $q=p$ and
$u(i)>u(i+1)$,}\end{array}\right.$

since $u\leqslant v$. Moreover $u_{l,i+1}=u_{l-1,i}\leqslant
z_{l-1,i}=z_{l,i+1}$ if $l\geqslant 2p+1$ while $u_{l,i+1}\leqslant
v_{l,i+1}=z_{l,i+1}$, if $l\leqslant 2p-2$. Let $l\in\\{2p-1,2p\\}$. There are
some cases to be considered.

  1. 1.

$q<p$ and $u(i)>u(i+1)$: in this case we have that
$u_{2p,i+1}=u_{2p-1,i}\leqslant z_{2p-1,i}=z_{2p,i+1}$ and so
$u_{2p-1,i+1}\leqslant u_{2p,i+1}-1\leqslant z_{2p,i+1}-1=z_{2p-1,i+1}$.

  2. 2.

$q<p$ and $u(i)<u(i+1)$: in this case we have that
$u_{2p-1,i+1}=u_{2p-1,i}-1\leqslant z_{2p-1,i}-1=z_{2p-1,i+1}$ and so
$u_{2p,i+1}=u_{2p-1,i+1}+1\leqslant z_{2p-1,i+1}+1=z_{2p,i+1}$.

  3. 3.

$q=p$ and $u(i)>u(i+1)$: in this case $u_{2p,i+1}=u_{2p-1,i}\leqslant
z_{2p-1,i}=z_{2p,i+1}$ and $u_{2p-1,i+1}=u_{2p-1,i}-1\leqslant
z_{2p-1,i}-1=z_{2p-1,i+1}$.

  4. 4.

$q=p$ and $u(i)<u(i+1)$: in this case $u_{2p-1,i+1}\leqslant
v_{2p-1,i+1}=a=z_{2p-1,i+1}$ and $u_{2p,i+1}=u_{2p-1,i+1}+1\leqslant
z_{2p-1,i+1}+1=z_{2p,i+1}$.

So we have proved that $u\leqslant z$ whenever $q\leqslant p$. Consider the
case $q>p$. If $l\leqslant 2p-2$ or $l\geqslant 2q$ the result follows as
above. Let $2p\leqslant l\leqslant 2q-2$. Then $u_{l,i}=u_{l,i+1}\leqslant
v_{l,i+1}=z_{l,i}$. Moreover

$u_{2p-1,i}=u_{2p-1,i+1}\leqslant v_{2p-1,i+1}=a<a+1=z_{2p-1,i}.$

Let $u(i)<u(i+1)$. Then

$u_{2q-1,i}=u_{2q-1,i+1}\leqslant v_{2q-1,i+1}=z_{2q-1,i}.$

If $u(i)>u(i+1)$ we have that $u_{2q-1,i}=u_{2q-1,i+1}+1\leqslant
v_{2q-1,i+1}+1=z_{2q-1,i}$ and then we have proved that $u\leqslant z$ also in
case $q>p$.

Let’s consider the case $s_{i+1}\in D(v)$. Define $a:=v(i+2)$ and
$v_{2p-1,i+1}:=a+1$. If $l\neq 2p-1$ we have that $u_{l,i+1}\leqslant
v_{l,i+1}=z_{l,i+1}$. Let $l=2p-1$. Since $u_{2p-1,i+1}\equiv
1\operatorname{(\mathrm{mod}~{}2)}$, $a\equiv
1\operatorname{(\mathrm{mod}~{}2)}$ and $u_{2p-1,i+1}\leqslant a+1$, we
conclude that $u_{2p-1,i+1}\leqslant a=z_{2p-1,i+1}$.

∎

######  Corollary 3.8.

Let $n\in\mathbb{P}$. Then $f_{n}:\operatorname{\mathcal{W}}(S_{n})\rightarrow
S_{\lfloor\frac{n}{2}\rfloor}$ is order preserving.

###### Proof.

If $n$ is even the result was already observed. If $n$ is odd let
$u,v\in\operatorname{\mathcal{W}}(S_{n})$ and $u\leqslant v$. We prove the
result by induction on $\operatorname{pos}(u,v)$. If
$\operatorname{pos}(u,v)=0$ by the tableau criterion we conclude that
$\chi_{n}(u)\leqslant\chi_{n}(v)$ and then the result follows by the even
case. Let $\operatorname{pos}(u,v)>0$ and $i:=\operatorname{pos}(v)$. In this
case, by Proposition 3.7, $u\leqslant v(i,i+2)<v$ or $u\leqslant
v(i+1,i+2)(i,i+2)<v(i+1,i+2)<v$. Hence, by the inductive hypothesis,
$f_{n}(u)\leqslant f_{n}(v(i,i+2))=f_{n}(v)$ or $f_{n}(u)\leqslant
f_{n}(v(i+1,i+2)(i,i+2))=f_{n}(v(i+1,i+2))=f_{n}(v)$. ∎

We can now prove the main result of this section.

######  Theorem 3.9.

Let $n>0$. Then $(\operatorname{\mathcal{W}}(S_{n}),\leqslant)$ is graded, of
rank ${{n}\choose{2}}-{{\lfloor\frac{n}{2}\rfloor}\choose{2}}$, and its rank
function is $\ell_{\operatorname{\mathcal{W}}}$.

###### Proof.

Assume first $n=2m$, $m>1$. Let $u,v\in\operatorname{\mathcal{W}}(S_{2m})$
with $u<v$, $\phi(u)=(\sigma,S)$ and $\phi(v)=(\tau,T)$. We prove that, if
$\ell_{\operatorname{\mathcal{W}}}(u,v)\neq 1$, then there exists
$z\in\operatorname{\mathcal{W}}(S_{2m})$ such that $u<z<v$ and
$\ell_{\operatorname{\mathcal{W}}}(z,v)=1$. Note that, by Proposition 3.2,
$\sigma\leqslant\tau$. We have two cases to consider.

  1. 1.

$\ell(\sigma,\tau)=0$: then $\sigma=\tau$ so, by Proposition 3.2, $S\subseteq
T$. Therefore there exists $i\in T\setminus S$. So $s_{2i-1}\in D(v)\setminus
D(u)$ hence, by the Lifting Property, $u<vs_{2i-1}<v$,
$vs_{2i-1}\in\operatorname{\mathcal{W}}(S_{2m})$ and
$\ell_{\operatorname{\mathcal{W}}}(vs_{i},v)=1$.

  2. 2.

$\ell(\sigma,\tau)>0$: in this case, since $\sigma<\tau$, there exist
$1\leqslant x<y\leqslant m$ such that $\sigma\leqslant\tau(x,y)\lhd\tau$. Let
$i:=2x-1$ and $j:=2y-1$. There are three cases to consider.

    1. (a)

$s_{i}\in D(v)$: let $z:=vs_{i}$ and $w:=(\tau(x,y),T)$. We want to prove that
$u<z$. Let $a+1:=v(i)=v_{2p-1,i}$, $b:=w(i)=v(j)=w_{2q-1,i}$ and
$c:=u(i)=u_{2s-1,i}$. In particular $q\leqslant p$. We only have to prove that
$u_{l,i}\leqslant(vs_{i})_{l,i}$ for all $l\leqslant i$. Since
$(vs_{i})_{l,i}=v_{l,i}$ if $l\neq 2p-1$, it is enough to prove that
$u_{2p-1,i}\leqslant(vs_{i})_{2p-1,i}$. Note that $u_{l,i+1}\leqslant
v_{l,i+1}=(vs_{i})_{l,i+1}$ for all $1\leqslant l\leqslant i+1$.

If $s\leqslant p$ then

$(vs_{i})_{2p-1,i}>(vs_{i})_{2p-2,i}=w_{2p-1,i}=w_{2p,i+1}\geqslant
u_{2p,i+1}\geqslant u_{2p-1,i}$

because $b<a$, and $\sigma\leqslant\tau(x,y)$.

If $s>p$ then
$u_{2p-1,i}=u_{2p-1,i+1}\leqslant(vs_{i})_{2p-1,i+1}=(vs_{i})_{2p-1,i}$. Since
$\ell_{\operatorname{\mathcal{W}}}(vs_{i},v)=1$ the result follows.

    2. (b)

$s_{i}\not\in D(v)$ and $s_{j}\in D(v)$: we then claim that $u<vs_{j}$. Let
$b+1=v(j)=v_{2p-1,j}$; then $b=(vs_{j})(j)=(vs_{j})_{2p-1,j}$. As in the case
above it is sufficient to prove that $u_{2p-1,j}\leqslant(vs_{j})_{2p-1,j}$.
If $u(j)>u_{2p-1,j}$ then $u_{2p-1,j}=u_{2p-1,j+1}=u_{2p,j+1}-1\leqslant
v_{2p,j+1}-1=b$. Let $u(j)\leqslant u_{2p-1,j}$; since
$\sigma\leqslant\tau(x,y)$ we have
$\\{u_{2p-1,j-1},u_{2p,j-1}\\}\leqslant\\{w_{2p-1,j-1},w_{2p,j-1}\\}$. Then,
$u_{2p-1,j}<u_{2p,j}=u_{2p-1,j-1}\leqslant w_{2p-1,j-1}=b$, since $a>b+1$.

    3. (c)

$s_{i},s_{j}\not\in D(v)$: in this case, by Lemma 3.5, $u<w^{A}_{x,y}(v)<v$
and $w^{A}_{x,y}(v)\in\operatorname{\mathcal{W}}(S_{2m})$,
$\ell_{\operatorname{\mathcal{W}}}(w^{A}_{x,y}(v),v)=1$.

Now assume $n=2m+1$, $m>0$. Let $u\leqslant v$, $i:=\operatorname{pos}(v)$,
and $\ell_{\operatorname{\mathcal{W}}}(u,v)>1$. We prove that there exists
$z\in\operatorname{\mathcal{W}}(S_{2m+1})$ such that $u<z<v$ and
$\ell_{\operatorname{\mathcal{W}}}(z,v)=1$. If $\operatorname{pos}(u,v)=0$
then the result follows by the previous point. If $\operatorname{pos}(u,v)>0$
we have, by Proposition 3.7, $u<v(i,i+2)<v$ if $s_{i+1}\not\in D(v)$ and
$u<vs_{i+1}<v$ otherwise. ∎

######  Remark 3.10.

In general $\operatorname{\mathcal{W}}(S_{n})\cap S_{n}^{J}$ is not graded;
one can see this by considering $J=\\{s_{1}\\}$ and the interval
$[124365,561234]$ in $\operatorname{\mathcal{W}}(S_{6})\cap S_{6}^{J}$.

We can now compute the rank-generating function of
$(\operatorname{\mathcal{W}}(S_{n}),\leqslant)$.

######  Corollary 3.11.

Let $m>0$. Then

$\operatorname{\mathcal{W}}(S_{2m})(x,\ell_{\operatorname{\mathcal{W}}})=(1+x)^{m}[m]_{x^{3}}!,$
$\operatorname{\mathcal{W}}(S_{2m+1})(x,\ell_{\operatorname{\mathcal{W}}})=(1+x)^{m}[m+1]_{x^{2}}[m]_{x^{3}}!.$

Moreover

$\operatorname{\mathcal{W}}(S_{2m})(x,\ell_{\operatorname{\mathcal{W}}})=\operatorname{\mathcal{W}}(S_{2m})(x,3\operatorname{emaj}+\operatorname{odes})$

and

$\operatorname{\mathcal{W}}(S_{2m+1})(x,\ell_{\operatorname{\mathcal{W}}})=\operatorname{\mathcal{W}}(S_{2m+1})(x,(3\operatorname{emaj}+\operatorname{odes})\circ\chi_{2m+1}+\operatorname{pos}).$

###### Proof.

The result follows by (1), Theorem 3.2 and the definition of
$\ell_{\operatorname{\mathcal{W}}}$. In fact,
$\operatorname{\mathcal{W}}(S_{2m+1})(x,\ell_{\operatorname{\mathcal{W}}})=[m+1]_{x^{2}}\operatorname{\mathcal{W}}(S_{2m})(x,\ell_{\operatorname{\mathcal{W}}})$
and $\ell(v)=\operatorname{odes}(v)+4\ell(f_{2m}(v))$, for all
$v\in\operatorname{\mathcal{W}}(S_{2m})$. ∎

From Proposition 3.11 we find that the polynomials
$\operatorname{\mathcal{W}}(S_{n})(x,\ell_{\operatorname{\mathcal{W}}})$ are
reciprocal, i.e. $x^{\ell_{\operatorname{\mathcal{W}}}(w_{n})}$
$\operatorname{\mathcal{W}}(S_{n})(x^{-1},\ell_{\operatorname{\mathcal{W}}})=\operatorname{\mathcal{W}}(S_{n})(x,\ell_{\operatorname{\mathcal{W}}})$.
In fact the poset $(\operatorname{\mathcal{W}}(S_{n}),\leqslant)$ is self-
dual, for all $n\in\mathbb{P}$, since the map $v\mapsto vw_{n}$ is an
antiautomorphism of $(\operatorname{\mathcal{W}}(S_{n}),\leqslant)$ such that
$\ell_{\operatorname{\mathcal{W}}}(vw_{n})=\ell_{\operatorname{\mathcal{W}}}(w_{n})-\ell_{\operatorname{\mathcal{W}}}(v)$
(see [2, Propositions 2.3.2 and 2.3.4]).

From the combinatorial description of the rank function of
$\operatorname{\mathcal{W}}(S_{2m+1})$ we can deduce a description of its
cover relations.

######  Proposition 3.12.

Let $m\in\mathbb{P}$, $u,v\in\operatorname{\mathcal{W}}(S_{2m+1})$,
$u=(i,\sigma,S)$ and $v=(j,\tau,T)$. Then $u\vartriangleleft v$ if and only if
either

  1. 1.

$i=j$, $\sigma=\tau$ and $S\vartriangleleft T$, or

  2. 2.

$j=i-1$, $\sigma=\tau$, $i-1\in S$ and $T=S\setminus\\{i-1\\}$, or

  3. 3.

$i=j$, $\sigma\vartriangleleft\tau$, $T\cap\\{a,b\\}=\varnothing$ and
$S=T\cup\\{a,b\\}$,

where $(a,b):=\tau^{-1}\sigma$. In particular, if
$u,v\in\operatorname{\mathcal{W}}(S_{2m})$, $u=(\sigma,S)$ and $v=(\tau,T)$,
then $u\vartriangleleft v$ if and only if either $\sigma=\tau$ and
$S\vartriangleleft T$, or $\sigma\vartriangleleft\tau$,
$T\cap\\{a,b\\}=\varnothing$ and $S=T\cup\\{a,b\\}$.

###### Proof.

If point $1$ or $2$ hold then
$\ell_{\operatorname{\mathcal{W}}}(v)-\ell_{\operatorname{\mathcal{W}}}(u)=1$
and by Theorem 3.4 there follows that $u\leqslant v$. If point $3$ holds then
the result follows by Lemma 3.6.

Conversely let $u\vartriangleleft v$. Then $\sigma\leqslant\tau$ by Corollary
3.8 and $\ell_{\operatorname{\mathcal{W}}}(u,v)=1$. If $i-j\geqslant 2$ then,
by Proposition 3.7 there is $z\in\operatorname{\mathcal{W}}(S_{2m+1})$ such
that $u\leqslant z<v$ and $\ell_{\operatorname{\mathcal{W}}}(z,v)=1$; so
$z=u$, which is a contradiction since $k-j\leqslant 1$, being $z=(k,\rho,R)$.
Hence assume $i-j\leqslant 1$.

If $\sigma<\tau$ then there exists a reflection $(r,s)$ such that
$\sigma\leqslant\tau(r,s)\vartriangleleft\tau$. If $i-j=1$ then by Proposition
3.7 $u=v(2j-1,2j+1)$ and $\sigma=\tau$, which is a contradiction. Therefore
$i=j$. As in the proof of Theorem 3.9, if $r\in T$ then
$(\sigma,S)\leqslant(\tau,T\setminus\\{r\\})$; if $r\not\in T$ and $s\in T$
then $(\sigma,S)\leqslant(\tau,T\setminus\\{s\\})$. Moreover
$\ell_{\operatorname{\mathcal{W}}}((\tau,T\setminus\\{r\\}),(\tau,T))=1$ and
$\ell_{\operatorname{\mathcal{W}}}((\tau,T\setminus\\{s\\}),(\tau,T))=1$ in
these cases. Then $(\sigma,S)=(\tau,T\setminus\\{r\\})$ or
$(\sigma,S)=(\tau,T\setminus\\{s\\})$, a contradiction. Therefore $r,s\not\in
T$ and, as in the proof of Theorem 3.9,
$(\sigma,S)\leqslant(\tau(r,s),T\cup\\{r,s\\})\vartriangleleft(\tau,T)$, and
$\ell_{\operatorname{\mathcal{W}}}((\tau(r,s),T\cup\\{r,s\\}),(\tau,T))=1$. So
we conclude that $S=T\cup\\{r,s\\}$ and
$\sigma=\tau(r,s)\vartriangleleft\tau$.

Assume now $\sigma=\tau$. If $i=j$ then $S\vartriangleleft T$ (else
$\ell_{\operatorname{\mathcal{W}}}(u,v)=\ell(u,v)\geqslant 2$). If $i-j=1$
then by Proposition 3.7 we have that $u=v(2j-1,2j+1)$ and $v<vs_{2j}$ and the
result follows.

The second statement follows from the first one by observing that
$\operatorname{\mathcal{W}}(S_{2m})$ is isomorphic to the interval
$[e,2m...321(2m+1)]$ in $\operatorname{\mathcal{W}}(S_{2m+1})$. ∎

For example the permutation $782156934\in\operatorname{\mathcal{W}}(S_{9})$
covers the Wachs permutations $781256934$, $782156439$ and $652187934$, which
correspond respectively to cases 1, 2 and 3 in Proposition 3.12.

From Proposition 3.12 we can now prove the second main result of this section,
namely a characterization of the Bruhat order relation on Wachs permutations.

######  Theorem 3.13.

Let $m>0$ and $u,v\in\operatorname{\mathcal{W}}(S_{2m+1})$, $u=(i,\sigma,S)$,
$v=(j,\tau,T)$. Then $u\leqslant v$ if and only if

$\sigma\leqslant\tau$, $S(u,v)\subseteq T(u,v)$, and $j\leqslant i$,

where, for $X\subseteq[m+1]$, $X(u,v):=X\cap([j-1]\cup[i,m])\cap
F(\sigma,\tau)$, being $F(\sigma,\tau):=\\{k\in[m]:\sigma(k)=\tau(k)\\}$.
Moreover $\ell_{\operatorname{\mathcal{W}}}(u)=3\ell(\sigma)+|S|+2(m-i+1)$. In
particular, if $u,v\in\operatorname{\mathcal{W}}(S_{2m})$, $u=(\sigma,S)$,
$v=(\tau,T)$, then $u\leqslant v$ if and only if $\sigma\leqslant\tau$ and
$S\cap F(\sigma,\tau)\subseteq T$, and
$\ell_{\operatorname{\mathcal{W}}}(u)=3\ell(\sigma)+|S|$.

###### Proof.

Let $u\leqslant v$. We may assume $u\vartriangleleft v$. There are three cases
to consider.

  1. 1.

$i=j$, $\sigma=\tau$ and $S\vartriangleleft T$: in this case
$F(\sigma,\tau)=[m]$ so the result follows.

  2. 2.

$j=i-1$, $\sigma=\tau$ and $S=T\cup\\{i-1\\}$, with $i-1\not\in T$: in this
case $F(\sigma,\tau)=[m]$ and $S\cap([i-2]\cup[i,m])\subseteq T$.

  3. 3.

$i=j$, $\sigma\vartriangleleft\tau$, $T\cap\\{a,b\\}=\varnothing$ and
$S=T\cup\\{a,b\\}$, where $(a,b)=\tau\sigma^{-1}$: we have
$F(\sigma,\tau)=[m]\setminus\\{a,b\\}$ and then $S\cap F(\sigma,\tau)\subseteq
T$.

Now let $\sigma\leqslant\tau$, $j<i$ and $S\cap([j-1]\cup[i,m])\cap
F(\sigma,\tau)\subseteq T$. In $S_{m}$ there exists a saturated chain
$\sigma=\sigma_{0}\vartriangleleft\sigma_{1}\vartriangleleft...\vartriangleleft\sigma_{n}=\tau$
with $n=\ell(\sigma,\tau)$. Define
$(a_{i},b_{i}):=\sigma^{-1}_{i}\sigma_{i-1}$ for all $i\in[n]$. We have the
following chain in $(\operatorname{\mathcal{W}}(S_{2m+1}),\leqslant)$:

$\displaystyle(i,\sigma,S)$ $\displaystyle\leqslant$
$\displaystyle(i,\sigma,S\cup[j,i-1])$ $\displaystyle\vartriangleleft$
$\displaystyle(i-1,\sigma,(S\cup[j,i-2])\setminus\\{i-1\\})$
$\displaystyle\vartriangleleft$
$\displaystyle(i-2,\sigma,(S\cup[j,i-3])\setminus\\{i-2,i-1\\})$
$\displaystyle\vartriangleleft...\vartriangleleft$
$\displaystyle(j+1,\sigma,(S\cup\\{j\\})\setminus[j+1,i-1])$
$\displaystyle\vartriangleleft$ $\displaystyle(j,\sigma,S\setminus[j,i-1])$
$\displaystyle\leqslant$
$\displaystyle(j,\sigma,(S\setminus[j,i-1])\cup\\{a_{1},b_{1}\\})$
$\displaystyle\vartriangleleft$
$\displaystyle(j,\sigma_{1},(S\setminus[j,i-1])\setminus\\{a_{1},b_{1}\\})$
$\displaystyle\leqslant$
$\displaystyle(j,\sigma_{1},(S\setminus([j,i-1]\cup\\{a_{1},b_{1}\\}))\cup\\{a_{2},b_{2}\\})$
$\displaystyle\vartriangleleft$
$\displaystyle(j,\sigma_{2},(S\setminus[j,i-1])\setminus\\{a_{1},b_{1},a_{2},b_{2}\\})$
$\displaystyle\leqslant...\leqslant$
$\displaystyle(j,\tau,(S\setminus[j,i-1])\setminus\\{a_{1},b_{1},...,a_{n},b_{n}\\})\leqslant(j,\tau,T),$

since $\\{a_{1},b_{1},...,a_{n},b_{n}\\}=[m]\setminus F(\sigma,\tau)$. The
length formula follows by Proposition 3.4.

The last statement follows immediately noting that the map
$(\sigma,S)\mapsto(m+1,\sigma,S)$ is a poset isomorphism between
$\operatorname{\mathcal{W}}(S_{2m})$ and
$\\{(i,\sigma,S)\in\operatorname{\mathcal{W}}(S_{2m+1}):i=m+1\\}$. ∎

We illustrate the preceding theorem with an example. Let
$u=(4,2431,\\{1,2,3\\})\in\operatorname{\mathcal{W}}(S_{9})$ and
$v=(3,3421,\\{2\\})\in\operatorname{\mathcal{W}}(S_{9})$. Then we have that
$2431<3421$ and $S\cap([j-1]\cup[i,m])\cap
F(\sigma,\tau)=\\{1,2,3\\}\cap(\\{1,2\\}\cup\\{4\\})\cap\\{2,4\\}=\\{2\\}$;
hence, by Theorem 3.13, $u<v$.

The following lemma can be easily deduced by Theorem 3.13 so we omit its
verification.

######  Lemma 3.14.

Let $m>0$ and $u,v\in\operatorname{\mathcal{W}}(S_{2m+1})$. If
$u\leqslant(i,\sigma,S_{1})\leqslant v$ and
$u\leqslant(i,\sigma,S_{2})\leqslant v$ in
$(\operatorname{\mathcal{W}}(S_{2m+1}),\leqslant)$ then
$u\leqslant(i,\sigma,S_{1}\cup S_{2})\leqslant v$.

The characterization obtained in Theorem 3.13 enables us to give an explicit
expression for the Möbius function of lower intervals in the poset of Wachs
permutations partially ordered by Bruhat order, and shows, in particular, that
it has values in $\\{0,1,-1\\}$.

######  Proposition 3.15.

Let $m>0$, and $v=(j,\tau,T)\in\operatorname{\mathcal{W}}(S_{2m+1})$. Then

$\mu(e,v)=\left\\{\begin{array}[]{ll}(-1)^{|T|},&\hbox{if $\tau=e$ and
$j=m+1$;}\\\ 0,&\hbox{otherwise.}\end{array}\right.$

In particular, if $v=(\tau,T)\in\operatorname{\mathcal{W}}(S_{2m})$ then

$\mu(e,v)=\left\\{\begin{array}[]{ll}(-1)^{|T|},&\hbox{if $\tau=e$;}\\\
0,&\hbox{otherwise.}\end{array}\right.$

###### Proof.

We proceed by induction on $\ell_{\operatorname{\mathcal{W}}}(v)$. If $\tau=e$
and $j=m+1$ then, by Theorem 3.13, the interval $[e,v]$ is isomorphic to a
Boolean algebra, so we conclude that $\mu(e,v)=(-1)^{|T|}$, as desired. So
assume that either $\tau\neq e$ or $j<m+1$. Then, by Proposition 3.12,
$\ell_{\operatorname{\mathcal{W}}}(v)\geqslant 2$, so, by Lemma 3.14 there
exists $R\subseteq[m]$, $R\neq\emptyset$, such that
$[e,v]\cap\\{(k,\rho,U)\in\operatorname{\mathcal{W}}(S_{2m+1}):\rho=e,k=m+1\\}=[e,(m+1,e,R)]$.
Hence

$\displaystyle\mu(u,v)$ $\displaystyle=$
$\displaystyle-\sum\limits_{x\in[e,v)}\mu(e,x)$ $\displaystyle=$
$\displaystyle-\sum\limits_{x\in[e,(i,\sigma,R)]}\mu(e,x)-\sum\limits_{x\in[e,v)\setminus[e,(i,\sigma,R)]}\mu(e,x)$
$\displaystyle=$
$\displaystyle-\sum\limits_{x\in[e,v)\setminus[e,(i,\sigma,R)]}\mu(e,x)=0$

by our induction hypothesis, and the fact that $|[e,(i,\sigma,R)]|\neq 1$,
where $[u,v):=\\{z\in\operatorname{\mathcal{W}}(S_{2m+1}):u\leqslant z<v\\}$.

The statement about Wachs permutations in the even case follows from the odd
one as in the proof of Theorem 3.13. ∎

We conclude by computing, using Proposition 3.15, the characteristic
polynomial of the poset of Wachs permutations.

######  Corollary 3.16.

The characteristic polynomial of
$(\operatorname{\mathcal{W}}(S_{n}),\leqslant)$ is

$(x-1)^{\left\lfloor\frac{n}{2}\right\rfloor}x^{\binom{n}{2}-\binom{\left\lfloor\frac{n}{2}\right\rfloor+1}{2}},$

for all $n\in\mathbb{P}$.

###### Proof.

The result follows from Poposition 3.15 and Theorem 3.9. ∎

The following is a commutative diagram that summarizes the poset morphisms
considered in this section. The function $\pi_{i}$ stands for the canonical
projection on the $i$-th factor of a Cartesian product. Notice that, if $A$
and $B$ are posets, the projection $\pi_{1}:A\otimes B\rightarrow A$ is order
preserving, whereas $\pi_{2}:A\otimes B\rightarrow B$ is not order preserving.

${{[m+1]^{*}}\times
S_{m}\times\mathcal{P}({[m]})}$${\operatorname{\mathcal{W}}(S_{2m+1})}$${\left({[m+1]^{*}}\times
S_{m}\right)\otimes\mathcal{P}({[m]})}$${S_{m}\times\mathcal{P}({[m]})}$${\operatorname{\mathcal{W}}(S_{2m})}$${S_{m}\otimes\mathcal{P}({[m]})}$${S_{m}}$$\scriptstyle{\phi^{-1}_{2m+1}}$$\scriptstyle{\pi_{2}\times\pi_{3}}$$\scriptstyle{\phi_{2m+1}}$$\scriptstyle{f_{2m+1}}$$\scriptstyle{\phi^{-1}_{2m}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\phi_{2m}}$$\scriptstyle{f_{2m}}$$\scriptstyle{\pi_{1}}$

##  4 Signed Wachs permutations and Bruhat order

For $n>0$ recall (see [4]) that the set of _signed Wachs permutation_ is

$\mathcal{W}(B_{n}):=\\{\sigma\in
B_{n}:|\sigma^{-1}(i)-\sigma^{-1}(i^{\ast})|\leqslant
1\;\forall\;i\in[n-1]\\}.$

So, for example, $[-2,-1,4,3]\in\mathcal{W}(B_{4})$ while
$[3,4,-2,1]\notin\mathcal{W}(B_{4})$. In the even case, as in type $A$, we
have the following group isomorphism (see Proposition 3.1)

$\operatorname{\mathcal{W}}(B_{2m})\simeq B_{m}\ltimes{\cal P}([m])=S_{2}\wr
B_{m}.$

We define a bijection $\phi:\mathcal{W}(B_{2m})\rightarrow B_{m}\times{\cal
P}([m])$ as follows. For $\sigma\in B_{m}$ and $T\subseteq[m]$ let
$\phi^{-1}(\sigma,T):=v$, where $v\in\mathcal{W}(B_{2m})$ is defined by

$v(2i-1)=\left\\{\begin{array}[]{ll}2\sigma(i)-\chi(\sigma(i)>0),&\hbox{if
$i\not\in T$,}\\\ 2\sigma(i)+\chi(\sigma(i)<0),&\hbox{if $i\in
T$,}\end{array}\right.$

and

$v(2i)=\left\\{\begin{array}[]{ll}2\sigma(i)+\chi(\sigma(i)<0),&\hbox{if
$i\not\in T$,}\\\ 2\sigma(i)-\chi(\sigma(i)>0),&\hbox{if $i\in
T$,}\end{array}\right.$

for all $i\in[m]$. For example, let $v:=[-3,-4,1,2,6,5]\in\mathcal{W}(B_{6})$;
then $\phi(v)=([-2,1,3],\\{1,3\\})$. Because of this bijection from now on we
freely identify the sets $\mathcal{W}(B_{2m})$ and $B_{m}\times{\cal P}([m])$,
so if $v\in\mathcal{W}(B_{2m})$ and $\phi(v)=(\sigma,T)$, then we simply write
$v=(\sigma,T)$ and we define

$\ell_{\mathcal{W}}(v):=\ell_{B}(v)-\ell_{B}(\sigma).$

Recall that we denote by $v\mapsto\tilde{v}$ the natural embedding
$B_{n}\hookrightarrow S_{\pm n}$. Note that if $v\in\mathcal{W}(B_{2m})$ then
$\tilde{v}\in\mathcal{W}(S_{\pm 2m})$. Indeed, if $v=(\sigma,T)$ then
$\tilde{v}=(\tilde{\sigma},-T\cup T)$. In fact, by Proposition 2.6, this is an
injective group and poset morphism
$\mathcal{W}(B_{2m})\hookrightarrow\mathcal{W}(S_{\pm 2m})$. For example, for
$u=[-2,-1,6,5,-3,-4]=([-1,3,-2],\\{2,3\\})\in\mathcal{W}(B_{6})$ we have

$\tilde{u}=(4,3,-5,-6,1,2,-2,-1,6,5,-3,-4)=((2,-3,1,-1,3,-2),\\{-3,-2,2,3\\})\in\mathcal{W}(S_{\pm
6}).$

Notice that if $n$ is odd then the image of a signed Wachs permutation is not
a Wachs permutation. For example, if
$u=[-2,-1,6,5,-3,-4,7]\in\mathcal{W}(B_{7})$ we have

$\tilde{u}=(-7,4,3,-5,-6,1,2,-2,-1,6,5,-3,-4,7)\not\in\mathcal{W}(S_{\pm 7}).$

It is known that $2\ell_{B}(v)=\ell_{A}(\tilde{v})+\operatorname{neg}(v)$ (see
e.g. [2, Exercise 8.2]) so we have that

$\ell_{\mathcal{W}}(v)=\frac{\ell_{\mathcal{W}}(\tilde{v})+\operatorname{neg}(\sigma)}{2},$
(6)

for all $v=(\sigma,T)\in\mathcal{W}(B_{2m})$, because
$2\operatorname{neg}(\sigma)=\operatorname{neg}(v)$.

######  Proposition 4.1.

The function $\operatorname{\mathcal{W}}(B_{2m})\rightarrow B_{m}$ defined by
the assignment $(\tau,T)\mapsto\tau$ is order preserving.

###### Proof.

Let $u=(\sigma,S)\in\operatorname{\mathcal{W}}(B_{2m})$ and
$v=(\tau,T)\in\operatorname{\mathcal{W}}(B_{2m})$. We have that $u\leqslant v$
implies $\tilde{u}\leqslant\tilde{v}$ and then, by Corollary 3.8,
$\tilde{\sigma}\leqslant\tilde{\tau}$. By Proposition 2.6, this implies
$\sigma\leqslant\tau$. ∎

It is easy to characterize, using Theorem 3.13, the Bruhat order relation
between signed Wachs permutations in the even case.

######  Proposition 4.2.

Let $u,v\in\mathcal{W}(B_{2m})$, $u=(\sigma,S)$, $v=(\tau,T)$. Then
$u\leqslant v$ if and only if $\sigma\leqslant\tau$ in $B_{m}$ and $S\cap
F(\sigma,\tau)\subseteq T$, where
$F(\sigma,\tau):=\\{i\in[m]:\sigma(i)=\tau(i)\\}$.

###### Proof.

We have that $\tilde{u}=(\tilde{\sigma},-S\cup S)$ and similarly
$\tilde{v}=(\tilde{\tau},-T\cup T)$. But, by Theorem 3.13,
$\tilde{u}\leqslant\tilde{v}$ if and only if
$\tilde{\sigma}\leqslant\tilde{\tau}$ and $(-S\cup S)\cap
F(\tilde{\sigma},\tilde{\tau})\subseteq-T\cup T$, which in turn happens if and
only if $\sigma\leqslant\tau$ and $S\cap F(\sigma,\tau)\subseteq T$. ∎

The next result is the analogue of Proposition 3.2.

######  Proposition 4.3.

Let $m>0$. Then

  1. 1.

$\phi:\mathcal{W}(B_{2m})\rightarrow B_{m}\otimes\mathcal{P}([m])$ is order
preserving;

  2. 2.

$\phi^{-1}:B_{m}\times\mathcal{P}([m])\rightarrow\mathcal{W}(B_{2m})$ is order
preserving.

Moreover $\ell_{B}(\tau,T)=4\ell_{B}(\tau)+|T|-\operatorname{neg}(\tau)$, for
all $(\tau,T)\in\mathcal{W}(B_{2m})$.

###### Proof.

Points 1. and 2. are direct consequences of Propositions 2.6 and 3.2. The last
equality follows by the formula of Proposition 3.2; in fact

$\displaystyle\ell_{B}(\tau,T)$ $\displaystyle=$
$\displaystyle\frac{\ell_{A}(\tilde{\tau},-T\cup
T)+\operatorname{neg}(\tau,T)}{2}=2\ell_{A}(\tilde{\tau})+|T|+\operatorname{neg}(\tau)$
$\displaystyle=$ $\displaystyle
4\ell_{B}(\tau)-2\operatorname{neg}(\tau)+|T|+\operatorname{neg}(\tau).$

∎

For any $(i,j)\in[n]\times[\pm n]$, $i\neq j$, we find it convenient to define

$(i,j)_{B}:=\left\\{\begin{array}[]{ll}(i,j)(-i,-j),&\hbox{if $i\neq|j|$;}\\\
(i,-i),&\hbox{otherwise.}\end{array}\right.$

So the set of reflections of $B_{n}$ is (see [2, Proposition 8.1.5])

$T^{B_{n}}=\\{(i,j)_{B}:1\leqslant i<|j|\leqslant
n\\}\cup\\{(i,-i)_{B}:i\in[n]\\}.$

Let $m>0$. For any reflection $(i,j)_{B}\in T^{B_{m}}$ we define an involution
$w^{B}_{i,j}:\mathcal{W}(B_{2m})\rightarrow\mathcal{W}(B_{2m})$ by letting

$w^{B}_{i,j}(\tau,T):=\left(\tau(i,j)_{B},T+\\{i,|j|\\}\right),$

for all $(\tau,T)\in\mathcal{W}(B_{2m})$, where $X+Y$ stands for the symmetric
difference between two sets $X$ and $Y$. For example, if
$v=([-2,1,4,3],\\{1,4\\})$ then $w^{B}_{3,-3}(v)=([-2,1,-4,3],\\{1,3,4\\})$
and $w^{B}_{1,-3}(v)=([-4,1,2,3],\\{3,4\\})$.

######  Remark 4.4.

We observe that, under the embedding
$\mathcal{W}(B_{2m})\hookrightarrow\mathcal{W}(S_{\pm 2m})$, we have that
$w^{B}_{i,j}(v)\mapsto w^{A}_{-i,-j}(w^{A}_{i,j}(\tilde{v}))$, if $i\neq-j$,
and $w^{B}_{i,-i}(v)\mapsto w^{A}_{i,-i}(\tilde{v})$.

The next technical result enables us to “lift” some order theoretic properties
from $B_{2m}$ to $\mathcal{W}(B_{2m})$. Its proof relies on the corresponding
result in type $A$, namely Lemma 3.5.

######  Lemma 4.5.

Let $m>0$, $i\in[m]$ and $j\in[\pm m]$. Assume
$v=(\tau,T)\in\mathcal{W}(B_{2m})$, $i,|j|\not\in T$ and
$\tau(i,j)_{B}\vartriangleleft\tau$. Then $w^{B}_{i,j}(v)<v$ and
$\ell_{\mathcal{W}}(w^{B}_{i,j}(v),v)=1$. Moreover if
$u=(\sigma,S)\in\mathcal{W}(B_{2m})$ and $\sigma\leqslant\tau(i,j)_{B}$ then
$u\leqslant w^{B}_{i,j}(v)<v$.

###### Proof.

Consider the element $\tilde{v}=(\tilde{\tau},-T\cup T)\in\mathcal{W}(S_{\pm
2m})$. Then by our hypothesis and Theorem 2.7 we have that
$\tilde{\tau}(i,j)(-i,-j)\vartriangleleft\tilde{\tau}(i,j)\vartriangleleft\tilde{\tau}$
if $i\neq|j|$, and $\tilde{\tau}(i,-i)\vartriangleleft\tilde{\tau}$ if
$i=|j|$. So, by Lemma 3.5, $\tilde{u}\leqslant
w^{A}_{-i,-j}(w^{A}_{i,j}(\tilde{v}))<w^{A}_{i,j}(\tilde{v})<\tilde{v}$ (since
$i,j\not\in T$ implies $-i,-j\not\in-T$) if $i\neq|j|$, and
$\tilde{u}\leqslant w^{A}_{i,-i}(\tilde{v})<\tilde{v}$ if $i=|j|$. By
Proposition 2.6 and Remark 4.4 we conclude that $u\leqslant w^{B}_{i,j}(v)<v$.
Note also that, by Theorem 2.7, $i\neq|j|$ and $v(i,j)_{B}\vartriangleleft v$
in $B_{n}$ imply $\operatorname{neg}(v(i,j)_{B})=\operatorname{neg}(v)$, and
that $v(i,-i)_{B}\vartriangleleft v$ in $B_{n}$ implies
$\operatorname{neg}(v(i,-i)_{B})=\operatorname{neg}(v)-1$. Therefore, if
$i,|j|\not\in T$ and $\tau(i,j)_{B}\vartriangleleft\tau$, then by Lemma 3.5
and (6) we have that $\ell_{\mathcal{W}}(w^{B}_{i,j}(v),v)=1$. ∎

The following result characterizes the cover relations of the ordering induced
by Bruhat order on the signed Wachs permutations in the even case.

######  Corollary 4.6.

Let $m>0$, $u,v\in\mathcal{W}(B_{2m})$, $u=(\sigma,S)$ and $v=(\tau,T)$. Then
$u\vartriangleleft v$ if and only if either one of the following conditions is
satisfied:

  1. 1.

$\sigma=\tau$ and $S\vartriangleleft T$;

  2. 2.

$\sigma\vartriangleleft\tau$, $T\cap\\{a,|b|\\}=\varnothing$ and
$S=T\cup\\{a,|b|\\}$, where $(a,b)_{B}:=\tau^{-1}\sigma$.

###### Proof.

Let $u\vartriangleleft v$ in $\operatorname{\mathcal{W}}(B_{2m})$. Then
$\sigma\leqslant\tau$ by Corollary 4.1. Moreover, if $\sigma<\omega<\tau$ for
some $\omega\in B_{m}$ then, by Proposition 4.3,
$u\leqslant(\omega,S)\leqslant v$. So $u\vartriangleleft v$ implies
$\sigma=\tau$ or $\sigma\vartriangleleft\tau$.

If $\sigma=\tau$ then, by Corollary 4.2, $S\subseteq T$; since
$u\vartriangleleft v$, we have that $S\vartriangleleft T$. Assume now
$\sigma\vartriangleleft\tau$ and let $(a,b)_{B}:=\tau^{-1}\sigma$. If $a\in T$
or $|b|\in T$ then, by point 2 of the proof of Theorem 3.9,
$\tilde{u}\leqslant\tilde{v}(a,a+1)(-a-1,-a)<\tilde{v}$ and
$\tilde{u}\leqslant\tilde{v}(|b|,|b|+1)(-|b|-1,-|b|)<\tilde{v}$, respectively.
Hence $u\vartriangleleft v$ and $\sigma\vartriangleleft\tau$ imply
$T\cap\\{a,|b|\\}=\varnothing$ and $u=w^{B}_{a,b}(v)$, by Lemma 4.5. This
implies $S=T\cup\\{a,|b|\\}$.

The converse can be proved by noting that if condition 1 or 2 are satisfied
then, by Proposition 3.12,
$[\tilde{u},\tilde{v}]=\\{\tilde{u},w_{1},w_{2},\tilde{v}\\}$, with
$w_{1},w_{2}\in\operatorname{\mathcal{W}}(S_{\pm
2m})\setminus\operatorname{\mathcal{W}}(B_{2m})$, or
$[\tilde{u},\tilde{v}]=\\{\tilde{u},\tilde{v}\\}$, where the intervals are
taken in $\operatorname{\mathcal{W}}(S_{\pm 2m})$. ∎

For example
$[-2,-1,3,4,6,5,-7,-8]\vartriangleleft[-2,-1,4,3,6,5,-7,-8]\vartriangleleft[-2,-1,5,6,3,4,-7,-8]$.
By Corollary 4.6 we have that if $u\vartriangleleft v$ holds in
$\mathcal{W}(B_{2m})$ then either $u=v(2i-1,2i)_{B}$ for some $i\in[m]$, or
$u=w^{B}_{i,j}(v)$ for some $(i,j)_{B}\in T^{B_{m}}$.

We can now prove the even part of the main result of this section.

######  Theorem 4.7.

The poset $\mathcal{W}(B_{2m})$ is graded, with rank function
$\ell_{\mathcal{W}}$, and its rank is $3m^{2}$.

###### Proof.

Since $e,w_{0}^{B_{2m}}\in\mathcal{W}(B_{2m})$, these are the minimum and
maximum of the poset $\mathcal{W}(B_{2m})$, respectively. Let
$u=(\sigma,S)\in\mathcal{W}(B_{2m})$ and $v=(\tau,T)\in\mathcal{W}(B_{2m})$ be
such that $u\vartriangleleft v$. Therefore, by Corollary 4.6, either
$u=v(2i-1,2i)_{B}$ for some $i\in[m]$ or $u=w^{B}_{i,j}(v)$, where
$\sigma=\tau(i,j)_{B}\vartriangleleft\tau$. In both cases
$\ell_{\mathcal{W}}(u,v)=1$, and this proves the first statement. The rank of
the poset $\operatorname{\mathcal{W}}(B_{2m})$ is, by (6),
$\ell_{\operatorname{\mathcal{W}}}(w_{0}(B_{2m}))=3m^{2}$, being
$\ell_{\mathcal{W}}\left(\widetilde{w_{0}(B_{2m})}\right)=m(6m-1)$. ∎

We now investigate the Bruhat order on $\operatorname{\mathcal{W}}(B_{n})$ for
$n$ odd. Let $v\in\operatorname{\mathcal{W}}(B_{2m+1})$. Note that there is a
bijection between $\operatorname{\mathcal{W}}(B_{2m+1})$ and
$\operatorname{\mathcal{W}}(B_{2m})\times[\pm(m+1)]$ given by
$v\mapsto(\overline{v},(\operatorname{pos}(v)+\operatorname{\mathrm{sgn}}(\operatorname{pos}(v)))/2)$
where

$\overline{v}(i)=\left\\{\begin{array}[]{ll}v(i),&\hbox{if
$i<v^{-1}(2m+1)$,}\\\ v(i+1),&\hbox{if $i\geqslant
v^{-1}(2m+1)$,}\end{array}\right.$

for $i\in[2m]$. Combining this with the bijection between
$\operatorname{\mathcal{W}}(B_{2m})$ and $B_{m}\times{\cal P}([m])$ explained
at the beginning of this section we obtain a bijection $\phi$ between
$\operatorname{\mathcal{W}}(B_{2m+1})$ and $[\pm(m+1)]\times B_{m}\times{\cal
P}([m])$. If $v\in\operatorname{\mathcal{W}}(B_{2m+1})$ and
$(i,\sigma,S)\in[\pm(m+1)]\times B_{m}\times{\cal P}([m])$ correspond under
this bijection then we write $v=(i,\sigma,S)$, and we define

$\ell_{W}(v):=\ell_{B}(v)-\ell_{B}(\sigma),$ (7)

and $\check{v}:=(\sigma,S)$ (so
$\check{v}\in\operatorname{\mathcal{W}}(B_{2m})$). So, for example, if
$v=[-1,-2,5,6,-7,3,4,]$ then $v=(-5,[-1,3,2],\\{-1\\})$ so
$\ell_{W}(v)=(9+3+7)-2=17$, and $\check{v}=[-1,-2,5,6,3,4]$. Note that, if
$u,v\in W(B_{2m+1})$ are such that $u^{-1}(2m+1)=v^{-1}(2m+1)$ then

$\ell_{W}(u,v)=\ell_{W}(\breve{u},\breve{v}).$ (8)

The next result is the analogue of Proposition 3.4.

######  Proposition 4.8.

Let $m>0$. Then

  1. 1.

$\phi:\mathcal{W}(B_{2m+1})\rightarrow[\pm(m+1)]^{*}\otimes
B_{m}\otimes\mathcal{P}([m])$ is order preserving;

  2. 2.

$\phi^{-1}:[\pm(m+1)]^{*}\times
B_{m}\times\mathcal{P}([m])\rightarrow\mathcal{W}(B_{2m+1})$ is order
preserving.

Moreover
$\ell_{B}(v)=4\ell_{B}(\tau)+|T|-\operatorname{neg}(\tau)+2(m-i+1)-3\chi(i<0)=\ell_{B}(\breve{v})+2(m-i+1)-3\chi(i<0)$,
for all $v=(i,\tau,T)\in\mathcal{W}(B_{2m+1})$.

###### Proof.

The first two points follow easily from Proposition 4.3 and the fact that if
$u\leqslant v$ then $\operatorname{pos}(u)\geqslant\operatorname{pos}(v)$. The
length equality is easy to check using our definitions and the well known fact
(see, e.g., [2, Prop. 8.1.1]) that
$\ell_{B}(v)=\operatorname{inv}(v)+\operatorname{neg}(v)+\operatorname{nsp}(v)$
for all $v$ in the hyperoctahedral group. ∎

For $v\in\operatorname{\mathcal{W}}(B_{2m+1})$ we define an element
$c(v)\in\operatorname{\mathcal{W}}(B_{2m+1})$ by

$c(v):=\left\\{\begin{array}[]{ll}v(-1,1)_{B},&\hbox{if $j=-1$;}\\\
v(j+1,j+2)_{B},&\hbox{if $j\neq-1$ and $v(j+1)>v(j+2)$;}\\\
v(j,j+2)_{B},&\hbox{if $j\neq-1$ and $v(j+1)<v(j+2)$;}\end{array}\right.$ (9)

where $j:=v^{-1}(2m+1)$. For example,
$c([-9,4,3,-6,-5,2,1,-8,-7])=[9,4,3,-6,-5,2,1,-8,-7]$,
$c([4,3,-6,-5,9,2,1,-8,-7])=[4,3,-6,-5,9,1,2,-8,-7]$ and
$c([3,4,-9,1,2,6,5,-7,-8])=[-9,4,3,1,2,6,5,-7,-8]$. Note that
$\ell_{\mathcal{W}}(c(v),v)=1$. The next result implies that $c(v)$ is the
only coatom $u$ of $v$ such that $v^{-1}(2m+1)\neq u^{-1}(2m+1)$, and is the
main technical tool in our proof of the fact that
$\operatorname{\mathcal{W}}(B_{n})$, under Bruhat order, is graded.

######  Theorem 4.9.

Let $m\in\mathbb{P}$, and $u,v\in D(B_{2m+1})$, $u<v$, be such that
$v^{-1}(2m+1)<u^{-1}(2m+1)$. Then $u\leqslant c(v)$.

###### Proof.

Let $j:=v^{-1}(2m+1)$. We distinguish various cases according as to whether
$j<-1$, $j=-1$, or $j>0$. Let, for brevity, $z:=c(v)$. Given a signed
permutation $w\in B_{n}$ and $j\in[\pm n]$ we find it convenient to denote by
$w_{j}$ the increasing rearrangement of $\\{w(-n),w(-n+1),\ldots,w(j)\\}$.

Assume first that $j>0$ and $j+1\in D_{R}(v)$. Let $a:=v(j+2)$ so
$v(j+1)=a+1$. Then $v_{k}=z_{k}$ for all
$k\in[\pm(2m+1)]\setminus\\{j+1,-j-2\\}$. Since $u,v,z$ are all Wachs
permutations there are
$x_{1},\ldots,x_{\tilde{m}},y_{1},\ldots,y_{\tilde{m}+1}\in[\pm(2m)]$, where
$\tilde{m}:=\frac{2m-1+j}{2}$, such that

$v_{j+1}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a+1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$z_{j+1}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$

and

$u_{j+1}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{\tilde{m}+1},y_{\tilde{m}+1}+1),$

for some $0\leqslant p\leqslant\tilde{m}$. Since $u_{j+1}\leqslant v_{j+1}$ we
have that $y_{p+1}\leqslant a+1$. But $y_{p+1}$ and $a$ are of the same parity
if they have the same sign so $y_{p+1}\leqslant a$ and hence $u_{j+1}\leqslant
z_{j+1}$.

Similarly, there are
$x_{1},\ldots,x_{\tilde{m}},y_{1},\ldots,y_{\tilde{m}}\in[\pm(2m)]$, where
$\tilde{m}:=\frac{2m-j-1}{2}$ such that

$v_{-j-2}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$
$z_{-j-2}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a-1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$

and

$u_{-j-2}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1),$

for some $0\leqslant p\leqslant\tilde{m}$. Since $u_{-j-2}\leqslant v_{-j-2}$
we have that $y_{p}\leqslant x_{p}+1$. But $x_{p}+1<-a-1$ so
$y_{p}+1\leqslant-a-1$ and hence $u_{-j-2}\leqslant z_{-j-2}$.

Suppose now that ($j>0$ and) $j+1\notin D_{R}(v)$. Consider first
$v_{j},z_{j},u_{j},v_{j+1},z_{j+1},u_{j+1}$. Since $v,u$, and $z$ are all
Wachs permutations, and by our definition of $z$, there are
$\\{x_{1},\ldots,x_{\tilde{m}}\\}_{<},$
$\\{y_{1},\ldots,y_{\tilde{m}}\\}_{<}\subseteq[\pm(2m)]$, where
$\tilde{m}:=\frac{2m-1+j}{2}$ such that

$v_{j}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$z_{j}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a+1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$
$u_{j}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{q},y_{q}+1,b^{\ast},y_{q+1},y_{q+1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$
$v_{j+1}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}+1},2m+1)$
$z_{j+1}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a,a+1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$
$u_{j+1}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{q},y_{q}+1,b,b+1,y_{q+1},y_{q+1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$

where $a:=v(j+1)$ ($=z(j+1)$), $b^{\ast}:=u(j)$, and $p,q\in[\tilde{m}]$. By
our hypothesis $u\leqslant v$ so $u_{j}\leqslant v_{j}$ and $u_{j+1}\leqslant
v_{j+1}$ (componentwise). This easily implies that $u_{j}\leqslant z_{j}$ and
$u_{j+1}\leqslant z_{j+1}$ (componentwise) keeping in mind the fact that
$x_{r}$ and $y_{s}$ have the same parity if they have the same sign for all
$r,s\in[\tilde{m}]$ (so $y_{s}\leqslant x_{r}+1$ implies $y_{s}\leqslant
x_{r}$ for all $r,s\in[\tilde{m}]$). For example, if $q<p$, then
$y_{p}\leqslant a$ (since $u_{j+1}\leqslant v_{j+1})$, while $y_{k}\leqslant
x_{k}+1$ (since $u_{j}\leqslant v_{j}$) so $y_{k}\leqslant x_{k}$ if
$p+1\leqslant k\leqslant\tilde{m}$. Similarly, if $q>p$, then
$y_{p+1}\leqslant a$ (since $u_{j+1}\leqslant v_{j+1}$) while $b\leqslant
x_{q+1}$ so $b\leqslant x_{q}$, while $y_{k}\leqslant x_{k}+1$ (since
$u_{j}\leqslant v_{j}$) so $y_{k}\leqslant x_{k}$ if $q+1\leqslant
k\leqslant\tilde{m}$, and $y_{k}\leqslant x_{k-1}+1$ (since $u_{j+1}\leqslant
v_{j+1}$) so $y_{k}\leqslant x_{k-1}$ if $p+2\leqslant k\leqslant q$. The case
$p=q$ is even simpler, and is therefore omitted.

Consider now $u_{-j-2},z_{-j-2},v_{-j-2}$ and
$u_{-j-1},z_{-j-1},v_{-j-1}$.Then reasoning as above we have that there are
$\\{x_{1},\ldots,x_{\tilde{m}}\\}_{<},\\{y_{1},\ldots,y_{\tilde{m}}\\}_{<}\subseteq[\pm(2m)]$
, where $\tilde{m}:=m-\frac{j+1}{2}$, such that

$v_{-j-2}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a-1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$
$z_{-j-2}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$
$u_{-j-2}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$
$v_{-j-1}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a-1,-a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$
$z_{-j-1}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$
$u_{-j-1}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{q},y_{q}+1,-b,y_{q+1},y_{q+1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$

where $p,q\in[\tilde{m}]$. As above, the fact that $u_{-j-2}\leqslant
v_{-j-2}$ and $u_{-j-1}\leqslant v_{-j-1}$ easily implies that
$u_{-j-2}\leqslant z_{-j-2}$ and $u_{-j-1}\leqslant z_{-j-1}$. For example, if
$q<p$, then $y_{k}\leqslant x_{k}+1$ (since $u_{-j-2}\leqslant v_{-j-2}$) so
$y_{k}\leqslant x_{k}$ for $1\leqslant k\leqslant p$ and hence
$-b<y_{q+1}\leqslant x_{q+1}$. Similarly, if $q\geqslant p$.

Consider now the case $j<-1$ (so $j\leqslant-3$). Assume first that $j+1\notin
D_{R}(v)$. Consider $u_{j},z_{j},v_{j},u_{j+1},z_{j+1},v_{j+1}$. If
$u^{-1}(-2m-1)>j$ then we conclude exactly as in the case $j>0$. So assume
that $u^{-1}(-2m-1)<j$. Then reasoning as above we conclude that there are
$\\{x_{1},\ldots,x_{\tilde{m}}\\}_{<}$,
$\\{y_{1},\ldots,y_{\tilde{m}}\\}_{<}\subseteq[\pm(2m)]$, where
$\tilde{m}:=m+\frac{j+1}{2}$, such that

$v_{j}=(x_{1},x_{1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$z_{j}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a+1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$
$u_{j}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$
$v_{j+1}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$z_{j+1}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a,a+1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1)$
$u_{j+1}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{q},y_{q}+1,c,y_{q+1},y_{q+1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$
$v_{j+2}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a,a+1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$u_{j+2}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{q},y_{q}+1,c,c+1,y_{q+1},y_{q+1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$

where $a:=v(j+1)$ ($=z(j+1)$), $c:=u(j+1)$, and $p,q\in[\tilde{m}]$. By our
hypothesis $u\leqslant v$ so $u_{j}\leqslant v_{j}$, $u_{j+1}\leqslant
v_{j+1}$, and $u_{j+2}\leqslant v_{j+2}$ (componentwise). As above, this
implies that $u_{j}\leqslant z_{j}$ and $u_{j+1}\leqslant z_{j+1}$. For
example, if $p<q$, then $y_{p+1}\leqslant a+1$ (so $y_{p+1}+1\leqslant
x_{p+1}$), $y_{p}+1\leqslant a$, $c\leqslant x_{q}+1$, and $y_{i}\leqslant
x_{i-1}+1$ (so $y_{i}\leqslant x_{i-1}$) for $p+2\leqslant i\leqslant q$
(since $u_{j+2}\leqslant v_{j+2}$), while $y_{i}\leqslant x_{i}+1$ (so
$y_{i}\leqslant x_{i}$) for $p+1\leqslant i\leqslant\tilde{m}$ ( since
$u_{j}\leqslant v_{j}$). Similarly, if $q<p$, then $y_{p}\leqslant a$ and
$y_{i}\leqslant x_{i}+1$ (so $y_{i}\leqslant x_{i}$) for $p+1\leqslant
i\leqslant\tilde{m}$ (since $u_{j+1}\leqslant v_{j+1}$). Finally, if $q=p$,
then $c\leqslant a+1$, $y_{p+1}\leqslant a$ and $y_{i}\leqslant x_{i}+1$ (so
$y_{i}\leqslant x_{i}$) for $p+1\leqslant i\leqslant\tilde{m}$ (since
$u_{j+2}\leqslant v_{j+2}$).

Consider now $u_{-j-1},v_{-j-1},z_{-j-1},u_{-j-2},v_{-j-2},z_{-j-2}$. If
$u^{-1}(-2m-1)>j$ then we conclude exactly as in the case $j>0$. So assume
that $u^{-1}(-2m-1)<j$. Then as above, since $u,v$, and $z$ are all Wachs
permutations we conclude that there are
$\\{x_{1},\ldots,x_{\tilde{m}}\\}_{<}\subseteq[\pm(2m)]$ and
$\\{y_{1},\ldots,y_{\tilde{m}}\\}_{<}\subseteq[\pm(2m)]$ such that

$v_{-j-2}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a-1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$z_{-j-2}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$u_{-j-2}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{q},y_{q}+1,c^{\ast},y_{q+1},y_{q+1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$
$v_{-j-1}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a-1,-a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$z_{-j-1}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$u_{-j-1}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{q},y_{q}+1,c,c+1,y_{q+1},y_{q+1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$
$v_{-j-3}=(x_{1},x_{1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
$u_{-j-3}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1)$

where $\tilde{m}:=m-\frac{j+3}{2}$, $-a:=v(-j-1)$ ($=z(-j-1)$), and
$c^{\ast}:=u(-j-2)$, for some $p,q\in[\tilde{m}]$. It is then not hard to
conclude that $u_{-j-2}\leqslant z_{-j-2}$ and $u_{-j-1}\leqslant z_{-j-1}$.
For example, if $p\leqslant q$, then $y_{i}\leqslant x_{i}+1$ (since
$u_{-j-2}\leqslant v_{-j-2}$) so $y_{i}\leqslant x_{i}$ for $1\leqslant
i\leqslant p$. If $p>q$ then $y_{i}\leqslant x_{i}+1$ ( since
$u_{-j-3}\leqslant v_{-j-3}$) so $y_{i}\leqslant x_{i}$ for $1\leqslant
i\leqslant\tilde{m}$ and hence $c+1<y_{q+1}\leqslant x_{q+1}$.

Assume now that ($j<-1$ and) $j+1\in D_{R}(v)$. Then $v_{k}=z_{k}$ if
$k\in[\pm(2m+1)]\setminus\\{-j-2,j+1\\}$. Assume first that
$u^{-1}(2m+1)\geqslant-j$. Let $a:=v(j+2)$ so $v(j+1)=a+1$. Then there are
$x_{1},\ldots,x_{\tilde{m}},y_{1},\ldots,y_{\tilde{m}}\in[\pm(2m)]$, where
$\tilde{m}:=\frac{2m-3-j}{2}$ such that

$v_{-j-2}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
(10)
$z_{-j-2}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,-a-1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
(11)

and

$u_{-j-2}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{q},y_{q}+1,c^{\ast},y_{q+1},y_{q+1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1),$

for some $0\leqslant p,q\leqslant\tilde{m}$ where $c^{\ast}:=u(-j-2)$. It is
then easy to see that $u_{-j-2}\leqslant z_{-j-2}$. Indeed, if $q<p$ then,
since $u_{-j-2}\leqslant v_{-j-2}$, $y_{p}\leqslant-a$ so, as above,
$y_{p}\leqslant-a-1$ and hence $u_{-j-2}\leqslant z_{-j-2}$. If $q\geqslant p$
then $y_{p}\leqslant x_{p}+1$ (since $u_{-j-2}\leqslant v_{-j-2}$) so
$y_{p}\leqslant x_{p}$. But $x_{p}+1<-a$ hence $y_{p}+1\leqslant-a-1$ so
$u_{-j-2}\leqslant z_{-j-2}$.

Similarly, there are
$\\{x_{1},\ldots,x_{\tilde{m}}\\}_{<},\\{y_{1},\ldots,y_{\tilde{m}}\\}_{<}\in[\pm(2m)]$,
where $\tilde{m}:=\frac{2m+j+1}{2}$, such that

$v_{j+1}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a+1,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
(12)
$z_{j+1}=(x_{1},x_{1}+1,\ldots,x_{p},x_{p}+1,a,x_{p+1},x_{p+1}+1,\ldots,x_{\tilde{m}},x_{\tilde{m}}+1,2m+1)$
(13)

and

$u_{j+1}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{q},y_{q}+1,c^{\ast},y_{q+1},y_{q+1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1),$

for some $0\leqslant p$, $q\leqslant\tilde{m}$, where $c^{\ast}:=u(j+1)$, and
we conclude exactly as in the last case.

Assume now that $u^{-1}(2m+1)<-j$. Then there are
$x_{1},\ldots,x_{\tilde{m}},y_{1},\ldots,y_{\tilde{m}+1}\in[\pm 2m]$, where
$\tilde{m}:=\frac{2m+1+j}{2}$ such that (12) and (13) hold and

$u_{j+1}=(y_{1},y_{1}+1,\ldots,y_{\tilde{m}+1},y_{\tilde{m}+1}+1)$

and the result follows easily. Similarly, there are
$x_{1},\ldots,x_{\tilde{m}},y_{1},\ldots,y_{\tilde{m}}\in[\pm 2m]$, where
$\tilde{m}:=\frac{2m-3-j}{2}$, such that (10) and (11) hold and

$u_{-j-2}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{\tilde{m}},y_{\tilde{m}}+1,2m+1)$

for some $0\leqslant p\leqslant\tilde{m}$. Hence, since $u_{-j-2}\leqslant
v_{-j-2}$, $y_{p}\leqslant x_{p}+1<-a-1$ so $y_{p}+1\leqslant-a-1$.

Finally, if $j=-1$ then $z=v(-1,1)$ so there are $x_{1},\ldots,x_{m}$,
$y_{1},\ldots,y_{m}\in[\pm 2m]$ such that
$v_{-1}=(x_{1},x_{1}+1,\ldots,x_{m},x_{m}+1,2m+1)$,
$z_{-1}=(-2m-1,x_{1},x_{1}+1,\ldots,x_{m},x_{m}+1)$ and
$u_{-1}=(-2m-1,y_{1},y_{1}+1,\ldots,y_{m},y_{m}+1)$, so the conclusion follows
easily. This concludes the proof. ∎

Recall that if $v=(\sigma,S,i)\in\operatorname{\mathcal{W}}(B_{2m+1})$ then we
let $\breve{v}=(\sigma,S)\in\operatorname{\mathcal{W}}(B_{2m})$. We can now
prove the main result of this section.

######  Theorem 4.10.

$\operatorname{\mathcal{W}}(B_{n})$ is graded, with rank function
$\ell_{\operatorname{\mathcal{W}}}$, and its rank is
$n^{2}-\left\lfloor\frac{n}{2}\right\rfloor^{2}$.

###### Proof.

If $n$ is even then this follows from Theorem 4.7. So assume that $n=2m+1$ for
some $m>0$. Since $\operatorname{\mathcal{W}}(B_{2m+1})$ has both a maximum
and a minimum element it is enough to show that if
$u,v\in\operatorname{\mathcal{W}}(B_{2m+1})$ and $u\lhd v$ then
$\ell_{\operatorname{\mathcal{W}}}(u,v)=1$. So let
$u,v\in\operatorname{\mathcal{W}}(B_{2m+1})$ be such that $u\lhd v$. Let
$j:=u^{-1}(2m+1)$, and $i:=v^{-1}(2m+1)$. Since $u<v$ we have that $i\leqslant
j$. If $i<j$ then by Theorem 4.9 we have that $u\leqslant c(v)<v$ so $u=c(v)$
and hence $\ell_{\operatorname{\mathcal{W}}}(u,v)=1$. If $i=j$ then
$\breve{u}\lhd\breve{v}$ so, by Theorem 4.7,
$\ell_{\operatorname{\mathcal{W}}}(\breve{u},\breve{v})=1$ and hence, by (8),
$\ell_{\operatorname{\mathcal{W}}}(u,v)=1$. Finally, it is not difficult to
see, by our definition of $\ell_{\operatorname{\mathcal{W}}}$, that
$\ell_{\operatorname{\mathcal{W}}}(w_{0}(B_{2m+1}))=3m^{2}+4m+1$. ∎

We remark that the sequence
$\\{\ell_{\operatorname{\mathcal{W}}}(w_{0}(B_{2m}))\\}_{m\in\mathbb{P}}$
gives the number of edges of the complete tripartite graph $K_{m,m,m}$ (see
A033428 in OEIS), and
$\\{\ell_{\operatorname{\mathcal{W}}}(w_{0}(B_{2m+1}))\\}_{m\in\mathbb{P}}$ is
the sequence of octagonal numbers (see A000567 in OEIS). We can now compute
the rank-generating function of
$(\operatorname{\mathcal{W}}(B_{n}),\leqslant)$.

######  Corollary 4.11.

Let $m>0$. Then

$\operatorname{\mathcal{W}}(B_{2m})(x,\ell_{\operatorname{\mathcal{W}}})=(1+x)^{m}[m]_{x^{3}}!\prod\limits_{i=1}^{m}(1+x^{3i-1})$

and

$\operatorname{\mathcal{W}}(B_{2m+1})(x,\ell_{\operatorname{\mathcal{W}}})=[m+1]_{x^{2}}(1+x^{2m+1})(1+x)^{m}[m]_{x^{3}}!\prod\limits_{i=1}^{m}(1+x^{3i-1}).$

###### Proof.

Note first that by our definition and Proposition 4.3 if
$u\in\operatorname{\mathcal{W}}(B_{2m})$, $u=(\sigma,S)$, then
$\ell_{\operatorname{\mathcal{W}}}(u)=3\ell_{B}(\sigma)+|S|-\operatorname{neg}(\sigma)$.
Therefore

$\sum_{u\in\operatorname{\mathcal{W}}(B_{2m})}x^{\ell_{\operatorname{\mathcal{W}}}(u)}=\sum_{\sigma\in
B_{m}}\sum_{S\subseteq[m]}x^{3\ell_{B}(\sigma)+|S|-\operatorname{neg}(\sigma)}=(1+x)^{m}\sum_{\sigma\in
B_{m}}x^{3\ell_{B}(\sigma)-\operatorname{neg}(\sigma)}.$

Let $J:=[m]$. Then by [2, Prop. 2.4.4] every element of $\sigma\in B_{m}$ may
be expressed in a unique way as $\sigma=zw$ where $w\in(B_{m})_{J}$ and
$z\in(B_{m})^{J}$. But the elements of $(B_{m})_{J}$ are permutations of
$S_{m}$ and the $z\in(B_{m})^{J}$ are characterized by the fact that
$z(1)<z(2)<\cdots<z(m)$. Hence these elements $z$ are in bijection with
subsets $S\subseteq[m]$, where the subset $S$ is the set of negative values
taken by $\sigma$. Furthermore, we then have that
$\ell_{B}(\sigma)=\operatorname{inv}(\sigma)+\operatorname{neg}(\sigma)+\operatorname{nsp}(\sigma)=\operatorname{inv}(w)+\operatorname{neg}(z)+\operatorname{nsp}(z)=\operatorname{inv}(w)+|S|+\sum_{s\in
S}(s-1)=\operatorname{inv}(w)+\sum_{s\in S}s$. We therefore have that

$\displaystyle\sum_{\sigma\in
B_{m}}x^{3\ell_{B}(\sigma)-\operatorname{neg}(\sigma)}=\sum_{w\in
S_{m}}\sum_{S\subseteq[m]}x^{3\operatorname{inv}(w)-|S|+3\sum_{s\in
S}s}=\sum_{w\in
S_{m}}x^{3\operatorname{inv}(w)}\sum_{S\subseteq[m]}x^{\sum_{s\in S}(3s-1)}$

and the first equation follows. For the second formula we have, using
Proposition 4.8 and (7),

$\displaystyle\sum_{u\in\operatorname{\mathcal{W}}(B_{2m+1})}x^{\ell_{\operatorname{\mathcal{W}}}(u)}$
$\displaystyle=$
$\displaystyle\left(\sum\limits_{i=-m-1}^{-1}x^{2m-2i-1}+\sum\limits_{i=1}^{m+1}x^{2(m-i+1)}\right)\operatorname{\mathcal{W}}(B_{2m})(x,\ell_{\operatorname{\mathcal{W}}})$
$\displaystyle=$
$\displaystyle[m+1]_{x^{2}}(1+x^{2m+1})\operatorname{\mathcal{W}}(B_{2m})(x,\ell_{\operatorname{\mathcal{W}}}).$

∎

Theorem 4.10 enables us to explicitly determine the cover relations in
$\operatorname{\mathcal{W}}(B_{n})$ for $n$ odd.

######  Corollary 4.12.

Let $u,v\in\operatorname{\mathcal{W}}(B_{2m+1})$, $u=(i,\sigma,S)$,
$v=(j,\tau,T)$ ($\sigma,\tau\in B_{m}$, $S,T\subseteq[m]$,
$i,j\in[\pm(m+1)]$). Then $u\lhd v$ if and only if either one of the following
conditions is satisfied:

  1. 1.

$i=j$, $\sigma=\tau$, and $S\lhd T$;

  2. 2.

$j=i-1$, $\sigma=\tau$ and either $|j+\chi(j<0)|\in S$,
$T=S\setminus\\{|j+\chi(j<0)|\\}$ if $j\neq-1$, or $S=T$ if $j=-1$;

  3. 3.

$i=j$, $\sigma\lhd\tau$, $T\cap\\{|a|,|b|\\}=\varnothing$ and
$S=T\cup\\{|a|,|b|\\}$, where $(a,b)_{B}=\tau\sigma^{-1}$.

###### Proof.

Note that, since $\operatorname{\mathcal{W}}(B_{2m+1})$ is graded, and its
rank function is $\ell_{\operatorname{\mathcal{W}}}$, $u\lhd v$ if and only if
$u\leqslant v$ and $\ell_{\operatorname{\mathcal{W}}}(u,v)=1$. If either 1. or
2. hold then it is easy to check that $u\leqslant v$ and $\ell_{W}(u,v)=1$.

Assume now that 3. holds. Suppose first that $\tau=\sigma(a,b)(-a,-b)$. Since
$\sigma\lhd\tau$ we have from Theorem 2.7 that $ab>0$. We may assume that
$0<a<b$. Again by Theorem 2.7 we have that
$\\{k\in[a+1,b-1]:\;\sigma(a)<\sigma(k)<\sigma(b)\\}=\varnothing$. This, by
3., implies that $\operatorname{inv}(v)-\operatorname{inv}(u)=4-2$,
$\operatorname{neg}(v)=\operatorname{neg}(u)$, and
$\operatorname{nsp}(v)=\operatorname{nsp}(u)$. Hence
$\ell_{\operatorname{\mathcal{W}}}(u,v)=\ell_{B}(u,v)-\ell_{B}(\sigma,\tau)=2-1=1$.
Suppose now that $\tau=\sigma(c,-c)$. We may clearly assume that $c>0$. Then
by Theorem 2.7 $\\{k\in[c-1]:\;-\sigma(c)<\sigma(k)<\sigma(c)\\}=\varnothing$.
This, again by 3., implies that
$\operatorname{inv}(v)-\operatorname{inv}(u)=-2(\sigma(c)-1)-1$,
$\operatorname{neg}(v)=\operatorname{neg}(u)+2$, and
$\operatorname{nsp}(v)=\operatorname{nsp}(u)+2(\sigma(c)-1)+1$. Hence
$\ell_{\operatorname{\mathcal{W}}}(u,v)=\ell_{B}(u,v)-\ell_{B}(\sigma,\tau)=2-1=1$.

Conversely, assume that $u\lhd v$. If $j>i$ then by Theorem 4.9 $u\leqslant
c(v)<v$ so $u=c(v)$. If $j=-1$ then 2. follows easily from (9). If $j\neq-1$
then $v(\operatorname{pos}(v)+1)<v(\operatorname{pos}(v)+2)$ (else, by (9),
$v^{-1}(2m+1)=c(v)^{-1}(2m+1)$ so $i=j$, a contradiction) and 2. again follows
from (9). If $i=j$ and $\sigma=\tau$ then it follows easily from Proposition
2.6 and Theorem 2.4 that $S\subseteq T$ and 1. follows since $u\lhd v$. So
assume ($i=j$ and) $\sigma<\tau$. Then, since $i=j$, $(\sigma,S)\lhd(\tau,T)$
in $\operatorname{\mathcal{W}}(B_{2m})$ so 3. follows from Corollary 4.6.

∎

So, for example, in $\operatorname{\mathcal{W}}(B_{9})$ we have that
$[9,2,1,-3,-4,8,7,-6,-5]\lhd[-9,2,1,-3,-4,8,7,-6,-5]$
$\lhd[1,2,-9,-3,-4,8,7,-6,-5]\lhd[1,2,-9,-3,-4,8,7,-5,-6]\lhd[1,2,-9,-6,-5,8,7,-4,-3]$,
where the first two covering relations are of type 2. (with $j=-1$, and
$j=-2$, respectively), the third one is of type 1., and the fourth one of type
3., with $(a,b)_{B}=(2,4)(-2,-4)$.

The next theorem characterizes the Bruhat order on signed Wachs permutations,
in the odd case. Note that it generalizes and puts in perspective the results
of Proposition 4.8.

######  Theorem 4.13.

Let $m>0$ and $u,v\in\operatorname{\mathcal{W}}(B_{2m+1})$, $u=(i,\sigma,S)$,
$v=(j,\tau,T)$. Then $u\leqslant v$ if and only if

$\mbox{$\sigma\leqslant\tau$, $\,$ $S(u,v)\subseteq T(u,v)$, and $\,$
$j\leqslant i$ },$

where, for $X\subseteq[m]$,
$X(u,v):=X\cap\left([\min\\{|i|,|j|\\}-1]\cup[\max\\{|i|,|j|\\},m]\right)\cap
F(\sigma,\tau)$, being $F(\sigma,\tau):=\\{i\in[m]:\sigma(i)=\tau(i)\\}$.
Moreover
$\ell_{\operatorname{\mathcal{W}}}(u)=3\ell_{B}(\sigma)+|S|-\operatorname{neg}(\sigma)+2(m-i+1)-3\chi(i<0)$.

###### Proof.

Let $u\leqslant v$. We may assume $u\vartriangleleft v$. It is clear from
Corollary 4.12 that $\sigma\leqslant\tau$ and $j\leqslant i$. Furthermore in
case 1 of Corollary 4.12 and in case 2 with $j=-1$ we have that $S\subseteq T$
and then $S(u,v)\subseteq T(u,v)$. In case 2 with $j\neq-1$ we have that $i$
and $j$ have the same sign and $F(\sigma,\tau)=[m]$; hence
$T=S\setminus\\{j\\}$ if $i>0$, so
$T(u,v)=\left(S\setminus\\{j\\}\right)\cap\left([j-1]\cup[j+1,m]\right)=S(u,v)$.
If $i<0$ then $T=S\setminus\\{|i|\\}$ and
$T(u,v)=\left(S\setminus\\{|i|\\}\right)\cap\left([|i|-1]\cup[|i|+1,m]\right)=S(u,v)$.
Finally in case 3 we have that $F(\sigma,\tau)=[m]\setminus\\{|a|,|b|\\}$;
since $S=T\cup\\{|a|,|b|\\}$ we obtain $T(u,v)=S(u,v)$.

Now let $\sigma\leqslant\tau$, $j\leqslant i$ and $S(u,v)\subseteq T(u,v)$.
Assume $j\leqslant i<0$. We then have the following chain in
$(\operatorname{\mathcal{W}}(B_{2m+1}),\leqslant)$:

$\displaystyle(i,\sigma,S)$ $\displaystyle\leqslant$
$\displaystyle(i,\sigma,S\cup[|i|,|j|-1])$ $\displaystyle\vartriangleleft$
$\displaystyle(i-1,\sigma,(S\cup[|i|+1,|j|-1])\setminus\\{|i|\\})$
$\displaystyle\vartriangleleft$
$\displaystyle(i-2,\sigma,(S\cup[|i|+2,|j|-1])\setminus\\{|i|,|i|+1\\})$
$\displaystyle\vartriangleleft...\vartriangleleft$
$\displaystyle(j,\sigma,S\setminus[|i|,|j|-1])\leqslant(j,\tau,T),$

where the last inequality follows from Proposition 4.2 and the facts that
$\left(S\setminus[|i|,|j|-1]\right)\cap F(\sigma,\tau)=S(u,v)\subseteq
T(u,v)\subseteq T$ and $(j,\sigma,S\setminus[|i|,|j|-1])\leqslant(j,\tau,T)$
if and only if $(\sigma,S\setminus[|i|,|j|-1])\leqslant(\tau,T)$. If $j<0<i$
and $|j|<i$ we have the following chain in
$(\operatorname{\mathcal{W}}(B_{2m+1}),\leqslant)$:

$\displaystyle(i,\sigma,S)$ $\displaystyle\leqslant$
$\displaystyle(i,\sigma,S\cup[|j|,i-1])$ $\displaystyle\vartriangleleft$
$\displaystyle(i-1,\sigma,(S\cup[|j|,i-2])\setminus\\{i-1\\})$
$\displaystyle\vartriangleleft$
$\displaystyle(i-2,\sigma,(S\cup[|j|,i-3])\setminus\\{i-2,i-1\\})$
$\displaystyle\vartriangleleft...\vartriangleleft$
$\displaystyle(-j,\sigma,S\setminus[|j|,i-1])\leqslant(j,\sigma,S\setminus[|j|,i-1])\leqslant(j,\tau,T),$

where the last inequality follows as in the previous case. If $j<0<i$ and
$|j|>i$ have that $(i,\sigma,S)\leqslant(-i,\sigma,S)\leqslant(j,\tau,T)$,
where the second inequality follows by the first case above. The case $i>j>0$
is similar and easier, so we omit it. The length formula follows from
Proposition 4.8. ∎

We illustrate the previous theorem with an example. Let $m=4$,
$u=[3,4,-5,-6,1,2,9,-7,-8]$ and $v=[-3,-4,-9,1,2,-5,-6,-8,-7]$. Then
$u=(4,[2,-3,1,-4],\\{2,4\\})$, and $v=(-2,[-2,1,-3,-4],$ $\\{1,3\\})$, so
$\sigma=[2,-3,1,-4]\leqslant[-2,1,-3,-4]=\tau$, $F(\sigma,\tau)=\\{4\\}$,
$j=-2\leqslant 4=i$, and $S(u,v)=\\{2,4\\}\cap\\{1,4\\}\cap\\{4\\}=\\{4\\}$,
$T(u,v)=\\{1,3\\}\cap\\{1,4\\}\cap\\{4\\}=\emptyset$, so $u\not\leqslant v$ in
$\operatorname{\mathcal{W}}(B_{9})$.

The following lemma is the analogue of Lemma 3.14 and can be easily deduced
from Theorem 4.13 so we omit its verification.

######  Lemma 4.14.

Let $m>0$ and $u,v\in\operatorname{\mathcal{W}}(B_{2m+1})$. If
$u\leqslant(i,\sigma,S_{1})\leqslant v$ and
$u\leqslant(i,\sigma,S_{2})\leqslant v$ in
$(\operatorname{\mathcal{W}}(B_{2m+1}),\leqslant)$ then
$u\leqslant(i,\sigma,S_{1}\cup S_{2})\leqslant v$.

The next result gives an explicit expression for the Möbius function of lower
intervals in the poset of signed Wachs permutations partially ordered by
Bruhat order, and shows, in particular, that it has values in $\\{0,1,-1\\}$.
The proof is similar to the one of Proposition 3.15 and we omit it.

######  Proposition 4.15.

Let $m>0$, and $v=(j,\tau,T)\in\operatorname{\mathcal{W}}(B_{2m+1})$. Then

$\mu(e,v)=\left\\{\begin{array}[]{ll}(-1)^{|T|},&\hbox{if $\tau=e$ and
$j=m+1$;}\\\ 0,&\hbox{otherwise.}\end{array}\right.$

In particular, if $v=(\tau,T)\in\operatorname{\mathcal{W}}(B_{2m})$ then

$\mu(e,v)=\left\\{\begin{array}[]{ll}(-1)^{|T|},&\hbox{if $\tau=e$;}\\\
0,&\hbox{otherwise.}\end{array}\right.$

By Proposition 4.15 and Theorem 4.10 we deduce the following result.

######  Corollary 4.16.

The characteristic polynomial of $\operatorname{\mathcal{W}}(B_{n})$ with the
Bruhat order is

$(x-1)^{\left\lfloor\frac{n}{2}\right\rfloor}x^{n^{2}-\left\lfloor\frac{n}{2}\right\rfloor^{2}-\left\lfloor\frac{n}{2}\right\rfloor},$

for all $n\in\mathbb{P}$.

${\displaystyle[-1,-2,-3]}$${\displaystyle[-2,-1,-3]}$${\displaystyle[2,1,-3]}$${\displaystyle[-3,-1,-2]}$${\displaystyle[1,2,-3]}$${\displaystyle[-3,-2,-1]}$${\displaystyle[3,-1,-2]}$${\displaystyle[-3,2,1]}$${\displaystyle[3,-2,-1]}$${\displaystyle[-3,1,2]}$${\displaystyle[3,2,1]}$${\displaystyle[-1,-2,3]}$${\displaystyle[3,1,2]}$${\displaystyle[-2,-1,3]}$${\displaystyle[2,1,3]}$${\displaystyle[1,2,3]}$
Figure 3: Hasse diagram of $(\operatorname{\mathcal{W}}(B_{3}),\leqslant)$.
${33}$${32}$${31}$${30}$${34}$${16}$${15}$${14}$${29}$${35}$${17}$${5}$${4}$${13}$${28}$3618631227${19}$${7}$12${11}$${26}$${20}$8${9}$10${25}$21${22}$${23}$24
Figure 4: Reading clockwise, the bold numbers on the diagonals of the hexagon
are the sequences
$\\{3,12,27,\ldots\\}=\\{\operatorname{\mathrm{rk}}(\operatorname{\mathcal{W}}(B_{2m}))\\}_{m>0}$,
$\\{2\operatorname{\mathrm{rk}}(\operatorname{\mathcal{W}}(S_{2m}))\\}_{m>0}$,
$\\{\operatorname{\mathrm{rk}}(\operatorname{\mathcal{W}}(B_{2m+1}))\\}_{m\geqslant
0}$ and
$\\{2\operatorname{\mathrm{rk}}(\operatorname{\mathcal{W}}(S_{2m+1}))\\}_{m>0}$.

##  5 Weak orders on Wachs permutations and signed Wachs permutations

In the previous sections we have proved several results concerning the Bruhat
order on Wachs permutations and signed Wachs permutations. The Bruhat order of
a Coxeter group is a refinement of two fundamental orders, the _left weak
order_ $\leqslant_{L}$ and the right one $\leqslant_{R}$ (see, e.g. [2,
Chapter 3]), whose Hasse diagrams are isomorphic to the Cayley graph of the
group, relative to the considered Coxeter presentation. Then it is natural to
ask when our results hold, and in what terms, for the left and right weak
orders on Wachs permutations and signed Wachs permutations.

Differently from the Bruhat order, for the right weak order the answer is
easily described as a Cartesian product (compare with Propositions 4.3 and
4.8). For reasons that will become clear in the next two pages we begin with
signed Wachs permutations. Recall that the set of reflections of $B_{n}$ is
$\\{(a,b)(-a,-b):1\leqslant a<|b|\leqslant n\\}\cup\\{(a,-a):1\leqslant
a\leqslant n\\}$. Therefore, if $v\in B_{n}$, $(a,b)(-a,-b)\in T_{L}(v)$ if
and only if $b>0$ and $b$ is to the left of $a$ in the complete notation of
$v$, or $b<0$ and $b$ is to the right of $a$, while $(a,-a)\in T_{L}(v)$ if
and only if $a$ is to the left of $-a$.

######  Theorem 5.1.

Let $n>0$; then
$(\operatorname{\mathcal{W}}(B_{n}),\leqslant_{R})\simeq\left(B_{\left\lceil\frac{n}{2}\right\rceil},\leqslant_{R}\right)\times\mathcal{P}\left(\left[\left\lfloor\frac{n}{2}\right\rfloor\right]\right)$.
In particular, $(\operatorname{\mathcal{W}}(B_{n}),\leqslant_{R})$ is a
complemented lattice.

###### Proof.

Let $n$ be even; for $v=(\tau,T)\in\operatorname{\mathcal{W}}(B_{n})$ we have
that

$\displaystyle T_{L}(\tau,T)$ $\displaystyle=$
$\displaystyle\biguplus\limits_{(a,-a)_{B}\in
T_{L}(\tau)}\left\\{(2a-1,-2a+1)_{B},(2a,-2a)_{B},(2a-1,-2a)_{B}\right\\}$
$\displaystyle\biguplus\limits_{(a,b)_{B}\in
T_{L}(\tau):b>0}\\{(2a-1,2b-1)_{B},(2a-1,2b)_{B},(2a,2b-1)_{B},(2a,2b)_{B}\\}$
$\displaystyle\biguplus\limits_{(a,b)_{B}\in
T_{L}(\tau):b<-a}\\{(2a-1,2b+1)_{B},(2a-1,2b)_{B},(2a,2b+1)_{B},(2a,2b)_{B}\\}$
$\displaystyle\uplus\left\\{(v(2a),v(2a-1))_{B}:a\in T,\,\tau(a)>0\right\\}$
$\displaystyle\uplus\left\\{(-v(2a-1),-v(2a))_{B}:a\in
T,\,\tau(a)<0\right\\}.$

Note that in the first group of reflections the non-symmetric ones (i.e., not
of the form $(k,-k)$ for some $k\in[n]$) are never simple and have odd first
element, and even and negative second one. The only simple reflections in the
second line have even first element and odd positive second one. The ones in
the third line are never simple, have negative second element, and if the
first one is odd and the second one even then the difference between the
absolute values of the two is $\geqslant 3$ (and hence are disjoint from the
non-symmetric ones in the first line). The reflections in the last two lines
are always simple, and have odd first element and even and positive second
one, since $(v(2a),v(2a-1))_{B}=(2\tau(a)-1,2\tau(a))_{B}$ if $\tau(a)>0$
while $(-v(2a-1),-v(2a))_{B}=(-2\tau(a)-1,-2\tau(a))_{B}$ if $\tau(a)<0$.

Therefore, if $u=(\sigma,S)\in\operatorname{\mathcal{W}}(B_{n})$ and
$v=(\tau,T)\in\operatorname{\mathcal{W}}(B_{n})$, we have that
$T_{L}(u)\subseteq T_{L}(v)$ if and only if $T_{L}(\sigma)\subseteq
T_{L}(\tau)$ and $\sigma_{S}\subseteq\tau_{T}$, where, for $X\subseteq[n/2]$
and $w\in B_{n/2}$, $w_{X}:=\left\\{w(i):i\in X\right\\}$. Indeed, if
$T_{L}(u)\subseteq T_{L}(v)$ and $(a,b)_{B}\in T_{L}(\sigma)$ is such that
(say) $b<-a$ then, by the equation written at the beginning of this proof,
$(2a,2b)_{B}\in T_{L}(u)$ so $(2a,2b)_{B}\in T_{L}(v)$ which implies, by the
remarks above, that there is $(c,d)_{B}\in T_{L}(\tau)$ such that $c<-d$ and
$(2a,2b)_{B}=(2c,2d)_{B}$, so $(a,b)_{B}=(c,d)_{B}\in T_{L}(\tau)$. Similarly
all the other cases. The converse is clear. Hence, by Proposition 2.3, the map
$(\tau,T)\mapsto(\tau,\tau_{T})$ gives the desired poset isomorphism.

Let $n$ be odd and $v=(j,\tau,T)\in\operatorname{\mathcal{W}}(B_{n})$. Then
similarly

$\displaystyle T_{L}(j,\tau,T)$ $\displaystyle=$
$\displaystyle\biguplus\limits_{(a,-a)_{B}\in T_{L}(\bar{\tau}):a\neq\lceil
n/2\rceil}\left\\{(2a-1,-2a+1)_{B},(2a,-2a)_{B},(2a-1,-2a)_{B}\right\\}$
$\displaystyle\biguplus\limits_{(a,b)_{B}\in T_{L}(\bar{\tau}):0<b<\lceil
n/2\rceil}\\{(2a-1,2b-1)_{B},(2a-1,2b)_{B},(2a,2b-1)_{B},(2a,2b)_{B}\\}$
$\displaystyle\biguplus\limits_{(a,b)_{B}\in T_{L}(\bar{\tau}):-\lceil
n/2\rceil<b<-a}\\{(2a-1,2b+1)_{B},(2a-1,2b)_{B},(2a,2b+1)_{B},(2a,2b)_{B}\\}$
$\displaystyle\biguplus\limits_{(a,b)_{B}\in T_{L}(\bar{\tau}):b=\lceil
n/2\rceil}\\{(2a-1,n)_{B},(2a,n)_{B}\\}$
$\displaystyle\biguplus\limits_{(a,b)_{B}\in T_{L}(\bar{\tau}):b=-\lceil
n/2\rceil,a\neq-b}\\{(2a-1,-n)_{B},(2a,-n)_{B}\\}$
$\displaystyle\uplus\\{(n,-n)_{B}:(\lceil n/2\rceil,-\lceil n/2\rceil)_{B}\in
T_{L}(\bar{\tau})\\}$
$\displaystyle\uplus\left\\{(2\tau(a)-1,2\tau(a))_{B}:a\in
T,\,\tau(a)>0\right\\}$
$\displaystyle\uplus\left\\{(2\tau(a)-1,2\tau(a))_{B}:-a\in
T,\,\tau(a)>0\right\\}$

where

$\bar{\tau}(k):=\left\\{\begin{array}[]{ll}\tau(k),&\hbox{if $k<|j|$;}\\\
\operatorname{\mathrm{sgn}}(j)\cdot(n+1)/2,&\hbox{if $k=|j|$;}\\\
\tau(k-1),&\hbox{if $k>|j|$},\\\ \end{array}\right.$

for all $k\in\lceil n/2\rceil$. The considerations made in the even case apply
line by line also to this case, with the added remark that the reflections in
lines 4, 5, and 6 are the only ones with $n$ or $-n$ in the second position.
Therefore, if $u=(i,\sigma,S)\in\operatorname{\mathcal{W}}(B_{n})$ and
$v=(j,\tau,T)\in\operatorname{\mathcal{W}}(B_{n})$, we have that
$T_{L}(u)\subseteq T_{L}(v)$ if and only if $T_{L}(\bar{\sigma})\subseteq
T_{L}(\bar{\tau})$ and $\sigma_{S}\subseteq\tau_{S}$, and we conclude as in
the previous case. The last statement follows from the fact that
$(B_{\left\lceil\frac{n}{2}\right\rceil},\leqslant_{R})$ is a complemented
lattice (see, e.g., [2, Cor. 3.2.2]). ∎

Regarding left weak order we have the following observations.

######  Proposition 5.2.

Let $m>0$. The posets $(\operatorname{\mathcal{W}}(B_{2m}),\leqslant_{R})$ and
$(\operatorname{\mathcal{W}}(B_{2m}),\leqslant_{L})$ are isomorphic.

###### Proof.

The assignment $v\mapsto v^{-1}$ defines a bijective function
$\operatorname{\mathcal{W}}(B_{2m})\rightarrow\operatorname{\mathcal{W}}(B_{2m})$
and $u\leqslant_{R}v$ if and ony if $u^{-1}\leqslant_{L}v^{-1}$, for all
$u,v\in B_{2m}$. ∎

On the other hand, the poset
$(\operatorname{\mathcal{W}}(B_{3}),\leqslant_{L})$ is not graded.

For Wachs permutations the situation is analogous but simpler so we leave to
the interested reader the proof of the following result.

######  Theorem 5.3.

Let $n>0$; then
$(\operatorname{\mathcal{W}}(S_{n}),\leqslant_{R})\simeq\left(S_{\left\lceil\frac{n}{2}\right\rceil},\leqslant_{R}\right)\times\mathcal{P}\left(\left[\left\lfloor\frac{n}{2}\right\rfloor\right]\right)$.
In particular, $(\operatorname{\mathcal{W}}(S_{n}),\leqslant_{R})$ is a
complemented lattice.

As in the case of signed Wachs permutations, the posets
$(\operatorname{\mathcal{W}}(S_{2m}),\leqslant_{R})$ and
$(\operatorname{\mathcal{W}}(S_{2m}),\leqslant_{L})$ are isomorphic, for all
$m>0$. On the other hand, the poset
$(\operatorname{\mathcal{W}}(S_{5}),\leqslant_{L})$ is not graded.

##  6 Open problems

In this section we collect some open problems and conjectures which arise in
this work, and the evidence that we have in their favor.

We have proved in Propositions 3.15 and 4.15 that the Möbius function of lower
intervals in the posets of Wachs permutations and signed Wachs permutations
partially ordered by Bruhat order always has values in $\\{0,1,-1\\}$. We feel
that this is true in general.

######  Conjecture 6.1.

Let $n\in\mathbb{P}$. Then

$\mu(u,v)\in\\{0,1,-1\\}$

for all $u,v\in\operatorname{\mathcal{W}}(S_{n})$.

We have verified Conjecture 6.1 for $n\leqslant 8$.

######  Conjecture 6.2.

Let $n\in\mathbb{P}$. Then

$\mu(u,v)\in\\{0,1,-1\\}$

for all $u,v\in\operatorname{\mathcal{W}}(B_{n})$.

We have verified Conjecture 6.2 for $n\leqslant 6$. Note that since
$\operatorname{\mathcal{W}}(S_{n})$ is isomorphic, as a poset, to the interval
$[e,[n,\ldots,3,2,1]]$ in $\operatorname{\mathcal{W}}(B_{n})$, Conjecture 6.2
implies Conjecture 6.1.

Recently Davis and Sagan [6] studied the convex hull of various sets of
pattern avoiding permutations. Following this idea, it is natural to look at
the convex hulls $c(\operatorname{\mathcal{W}}(S_{n}))$ and
$c(\operatorname{\mathcal{W}}(B_{n}))$ of Wachs and signed Wachs permutations
in $\operatorname{\mathbb{R}}^{n}$. In this respect, we feel that the
following is true.

######  Conjecture 6.3.

Let $m\in\mathbb{P}$. Then $c(\operatorname{\mathcal{W}}(S_{2m}))$ is a simple
polytope.

We have verified Conjecture 6.3 for $m\leqslant 5$. According to SageMath [21]
$c(\operatorname{\mathcal{W}}(S_{9}))$ is not simple.

######  Conjecture 6.4.

Let $m\in\mathbb{P}$. Then $c(\operatorname{\mathcal{W}}(B_{2m}))$ is a simple
polytope.

We have verified Conjecture 6.4 for $m\leqslant 3$. According to SageMath [21]
$c(\operatorname{\mathcal{W}}(B_{3}))$ is not simple.

Regarding left weak order, we feel that the following might be true.

######  Problem 1.

Is $(\operatorname{\mathcal{W}}(S_{2m+1}),\leqslant_{L})$ a lattice for all
$m\in\mathbb{P}$?

We have verified that the answer to Problem 1 is yes if $m\leqslant 4$.

Acknowledgments. The first named author was partially supported by the MIUR
Excellence Department Project CUP E83C18000100006.

## References

  * [1] S. Billey, V. Lakshmibai, Singular Loci of Schubert Varieties, Progress in Mathematics, 182, Birkhäuser Boston Inc., Boston, MA, 2000, xii+251.
  * [2] A. Björner, F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231, Springer-Verlag, New York, 2005.
  * [3] A. Björner, M. Wachs, Generalized quotients in Coxeter groups, Trans. Amer. Math. Soc. 308 (1988), 1–37.
  * [4] F. Brenti and P. Sentinelli, Odd and even major indices and one-dimensional characters for classical Weyl groups, Annals of Combinatorics 24 (4), 809-835 (2020)
  * [5] M. Can, Y. Cherniavsky, T. Twelbeck, Lexicographic shellability of the Bruhat–Chevalley order on fixed-point-free involutions, Israel J. Math., 207(2015), 281–299.
  * [6] R. Davis and B. Sagan, Pattern-avoiding polytopes, Europ. J. Combin., 74, 48-84 (2018).
  * [7] W. Fulton, Young tableaux. With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts 35, Cambridge University Press, Cambridge, 1997.
  * [8] M. Hansson, The Bruhat order on conjugation-invariant sets of involutions in the symmetric group, J. Algebraic Combin., 44 (2016), 849–862.
  * [9] A. Hultman, Fixed points of involutive automorphisms of the Bruhat order, Adv. Math., 195 (2005), 283–296.
  * [10] A. Hultman, Twisted identities in Coxeter groups, J. Algebraic Combin., 28 (2008), 313–332.
  * [11] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, no.29, Cambridge Univ. Press, Cambridge, 1990.
  * [12] F. Incitti, The Bruhat order on the involutions of the hyperoctahedral group, European J. Combin. 24 (2003), 825–848.
  * [13] F. Incitti, The Bruhat order on the involutions of the symmetric group, J. Algebraic Combin. 20 (2004), 243–261.
  * [14] F. Incitti, Bruhat order on classical Weyl groups: minimal chains and covering relation, European J. Combin., 26 (2005), 729-753.
  * [15] F. Incitti, Bruhat order on the involutions of classical Weyl groups, Adv. in Appl. Math., 37 (2006), 68–111.
  * [16] I. G. Macdonald, Notes on Schubert Polynomials, Publ. du LACIM 6, University du Québec, Montréal, 1991.
  * [17] L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. viii+167 pp.
  * [18] R. Richardson, T. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), 389–436; and 49 (1994), 231–238.
  * [19] P. Sentinelli, Complements of Coxeter group quotients, J. Algebraic Combin., 41 (2015), 727–750.
  * [20] R. P. Stanley, Enumerative Combinatorics , vol.1, Second Edition, Cambridge Studies in Advanced Mathematics, no.49, Cambridge Univ. Press, Cambridge, 2012.
  * [21] The Sage Developers, _SageMath, the Sage Mathematics Software System (Version 8.6)_ , 2019, https://www.sagemath.org.
  * [22] M. L. Wachs, An involution for signed Eulerian numbers, Discrete mathematics 99.1-3 (1992): 59-62.