#  Refinements of Beck-type partition identities

T. Amdeberhan  Department of Mathematics
Tulane University
New Orleans, LA 70118, USA<EMAIL_ADDRESS>,  G. E. Andrews  Department
of Mathematics
Penn State University
University Park, PA 16802, USA<EMAIL_ADDRESS>and  C. Ballantine  Department of
Mathematics and Computer Science
College of the Holy Cross
Worcester, MA 01610, USA
<EMAIL_ADDRESS>
###### Abstract.

Franklin’s identity generalizes Euler’s identity and states that the number of
partitions of $n$ with $j$ different parts divisible by $r$ equals the number
of partitions of $n$ with $j$ repeated parts. In this article, we give a
refinement of Franklin’s identity when $j=1$. We prove Franklin’s identity
when $j=1$, $r=2$ for partitions with fixed perimeter, i.e., fixed largest
hook. We also derive a Beck-type identity for partitions with fixed perimeter:
the excess in the number of parts in all partitions into odd parts with
perimeter $M$ over the number of parts in all partitions into distinct parts
with perimeter $M$ equals the number of partitions with perimeter $M$ whose
set of even parts is a singleton. We provide analytic and combinatorial proofs
of our results.

##  1\. Introduction

A partition of $n\in\mathbb{N}$ is a finite non-increasing sequence of
positive integers that add up to $n$. We use the notation
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{\ell})$, with
$\lambda_{1}\geq\lambda_{2}\geq\ldots,\lambda_{\ell}$ and
$\lambda_{1}+\lambda_{2}+\cdots+\lambda_{\ell}=n$. We refer to the numbers
$\lambda_{i}$ as the parts of $\lambda$. The size of $\lambda$, denoted by
$|\lambda|$, is the sum of all its parts. We write $\lambda\vdash n$ to mean
$\lambda$ is a partition of size $n$. The length of $\lambda$, denoted by
$\ell(\lambda)$, is the number of parts of $\lambda$. For a non-negative
integer $n$, we denote by $p(n)$ the number of partitions of $n$. Moreover, we
write $p(n\mid X)$ for the number of partitions of $n$ satisfying condition
$X$. For more details about partitions, we refer the reader to [1].

A classical partition identity is an identity of the form $p(n\mid X)=p(n\mid
Y)$. One of the oldest partition identity is Euler’s identity

$p(n\mid\text{parts are odd})=p(n\mid\text{parts are distinct}).$

Glaisher’s generalization of Euler’s identity states that, for $r\geq 2$,

$p(n\mid\text{parts are not divisible by $r$})=p(n\mid\text{parts are repeated
less than $r$ times}).$

We note that partitions with parts not divisible by $r$ are called $r$-regular
partitions. Franklin generalized Glaisher’s identity further. To state the
identity, we denote by $\mathcal{O}(n;r,j)$ the set of partitions of $n$ in
which exactly $j$ different parts are divisible by $r$ and these parts can be
repeated. We also denote by $\mathcal{D}(n;r,j)$ the set of partitions of $n$
in which exactly $j$ parts occur at least $r$ times. Then, Franklin’s identity
states that, for $r\geq 2$ and any non-negative integer $j$,

$|\mathcal{O}(n;r,j)|=|\mathcal{D}(n;r,j)|.$

For additional discussions on partitions and partition identities, again refer
to [1].

In this article, we prove a refinement of Franklin’s identity for $j=1$. For a
positive integer $u$, we denote by $\mathcal{O}_{u}(n;r,1)$ the set of
partitions $\lambda\in\mathcal{O}(n;r,1)$ such that the unique part of
$\lambda$ divisible by $r$ is repeated exactly $u$ times. Moreover, we denote
by $\mathcal{D}_{u}(n;r,1)$ the set of partitions
$\lambda\in\mathcal{D}(n;r,1)$ such that the unique part occuring at least $r$
times equals $u$. Let $\alpha_{u}^{(r)}(u)=|\mathcal{O}_{u}(n;r,1)|$ and
$\beta^{(r)}(u)=|\mathcal{D}_{u}(n;r,1)|$.

######  Theorem 1.1.

Let $u,r$ be integers such that $u\geq 1$ and $r\geq 2$. Then, for any non-
negative integer $n$, we have

$\alpha_{u}^{(r)}(n)=\beta_{u}^{(r)}(n).$

In 2017, Beck [4] conjectured and Andrews [2] proved a companion identity to
Euler’s identity relating the number of parts in all partitions involved in
Euler’s identity. Denote by $a(n)$ the number of parts in all partitions in
$\mathcal{O}(n;2,0)$, i.e., partitions of $n$ into odd parts, and by $b(n)$
the number of parts in all partitions in $\mathcal{D}(n;2,0)$, i.e.,
partitions of $n$ into distinct parts.

######  Theorem 1.2 (Andrews).

If $n$ is a non-negative integer, we have

$a(n)-b(n)=|\mathcal{O}(n;2,1)|=|\mathcal{D}(n;2,1)|.$

A Beck-type companion identity to Glaisher’s identity was provided by Yang
[9]. Ballantine and Welch [3] gave Beck-type companion identities to
Franklin’s identity.

In [8], Straub discovered that Euler’s identity holds for partitions of fixed
perimeter rather than fixed size. In [6], the perimeter of a partition
$\lambda$ is defined to be the maximum hook length in the Ferrers diagram of
$\lambda$, i.e., $\lambda_{1}+\ell(\lambda)-1$.

######  Theorem 1.3 (Straub).

The number of partitions into distinct parts with perimeter $M$ is equal to
the number of partitions into odd parts with perimeter $M$. Both are
enumerated by the Fibonacci number $F_{M}$.

Motivated by Straub’s theorem and the Beck-type companion identities toward
generalizations of Euler’s theorem, we study several partition numbers for
certain partitions of fixed perimeter and for the number of parts in these
sets of partitions.

First, let’s fix some notations. We denote $\mathcal{G}(M)$ (respectively
$\mathcal{H}(M)$) the set of all partitions into odd parts (respectively
distinct parts) with perimeter $M$. Let $g(M)$ be the total number parts in
all partitions in $\mathcal{G}(M)$, and $h(M)$ be the total number of parts in
all partitions in $\mathcal{H}(M)$. Moreover, we denote by
$\mathcal{G}_{1}(M)$ the set of partitions with perimeter $M$ in which all
parts are odd except one even part (possibly repeated). We also denote by
$\mathcal{H}_{1}(M)$ the set of partitions with perimeter $M$ where one part
occurs at least twice and all other parts occur once. We let
$g_{1}(M)=|\mathcal{G}_{1}(M)|$ and $h_{1}(M)=|\mathcal{H}_{1}(M)|$.

######  Example 1.

We list the first few terms of each sequence starting with $M=1$.

  1. (1)

$g(M)$ starts with: $1,2,4,8,15,28,51,92,\dots$.

  2. (2)

$h(M)$ starts with: $1,1,3,5,10,18,33,59,\dots$.

  3. (3)

$g_{1}(M)$ starts with: $0,1,2,5,10,20,38,71,\dots$.

  4. (4)

$h_{1}(M)$ starts with: $0,1,2,5,10,20,38,71,\dots$.

  5. (5)

$g(M)-h(M)$ starts with: $0,1,1,3,5,10,18,33,\dots$.

The generating functions for these sequences are given in the next theorem and
its corollary.

######  Theorem 1.4.

The generating functions for the sequences $g(M),h(M),g_{1}(M)$, and
$h_{1}(M)$ are

$\displaystyle\sum_{M\geq 1}g(M)\,x^{M}$
$\displaystyle=\frac{x-x^{3}}{(1-x-x^{2})^{2}},\qquad\sum_{M\geq
1}h(M)\,x^{M}=\frac{x-x^{2}}{(1-x-x^{2})^{2}},$ $\displaystyle\sum_{M\geq
1}g_{1}(M)\,x^{M}$ $\displaystyle=\frac{x^{2}}{(1-x-x^{2})^{2}}=\sum_{M\geq
1}h_{1}(M)\,x^{M}.$

In the proof of this theorem, we also furnish formulas for these sequences.

######  Corollary 1.5.

$\displaystyle\sum_{M\geq 1}(g_{1}(M)-g_{1}(M-1))\,x^{M}$
$\displaystyle=\frac{x^{2}-x^{3}}{(1-x-x^{2})^{2}}=\sum_{M\geq
1}(h_{1}(M)-h_{1}(M-1))\,x^{M},$ $\displaystyle\sum_{M\geq
1}(g(M)-h(M))\,x^{M}$ $\displaystyle=\frac{x^{2}-x^{3}}{(1-x-x^{2})^{2}}.$

Next, we provide recurrence relations for $g(M),h(M),g_{1}(M)$, and
$h_{1}(M)$.

######  Theorem 1.6.

For all $M\geq 2$, we have

(1) $\displaystyle g(M)$ $\displaystyle=g(M-1)+g(M-2)+F_{M-1},$ (2)
$\displaystyle h(M)$ $\displaystyle=h(M-1)+h(M-2)+F_{M-2},$ (3) $\displaystyle
g_{1}(M)$ $\displaystyle=g_{1}(M-1)+g_{1}(M-2)+F_{M-1},$ (4) $\displaystyle
h_{1}(M)$ $\displaystyle=h_{1}(M-1)+h_{1}(M-2)+F_{M-1}.$

The next corollary follows immediately from Theorem 1.6. It can also be proven
by making use of the generating functions in Theorem 1.4. We adopt the
convention that $F_{-1}=1,F_{0}=0,F_{1}=1$.

######  Corollary 1.7.

For $M\geq 2$, we have

  1. (1)

$g(M)=F_{M}+\sum_{k=1}^{M-1}F_{k}F_{M-k}$;

  2. (2)

$h(M)=F_{M}+\sum_{k=1}^{M-2}F_{k}F_{M-1-k}$;

  3. (3)

$g_{1}(M)=h_{1}(M)=\sum_{k=1}^{M-1}F_{k}F_{M-k}$;

  4. (4)

$g(M)-h(M)=g_{1}(M)-g_{1}(M-1)$

$=h_{1}(M)-h_{1}(M-1)=F_{M-1}+\sum_{k=1}^{M-3}F_{k}F_{M-2-k}.$

Part (4) of Corollary 1.7 leads to a Beck-type companion identity to Straub’s
theorem.

######  Corollary 1.8.

$g(M)-h(M)$ equals the number of partitions in $\mathcal{G}_{1}(M)$ with no
part equal to $1$.

######  Remark 1.

The identity $g_{1}(M)=h_{1}(M)$ in part (3) of Corollary 1.7 is Franklin’s
identity for $j=1$, $r=2$ applied to partitions of fixed perimeter.

##  2\. Partitions with fixed perimeter

The Ferrers diagram of a partition
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell})$ is an array of left-
justified boxes such that the $i$-th row from the top contains $\lambda_{i}$
boxes. For example, the Ferrers diagram of the partition
$\lambda=(6,6,3,2,2,1)$ is shown below.

$\tiny\ydiagram{6,6,3,2,2,1}$

We can also describe a partition $\lambda$ by a binary string or bit string
$b(\lambda)$ starting with $1$ and ending with $0$ as follows. Start at SW
corner of the Ferrers diagram and travel along the outer profile going NE. The
profile is a Dyck path. For each step to the right, record a $1$; for each
step up, record a $0$. For example, the bit string representing the partition
$\lambda=(6,6,3,2,2,1)$ is $b(\lambda)=101001011100$. The perimeter of the
partition $\lambda$ is one less than the number of digits in $b(\lambda)$. The
number of parts of $\lambda$ equals the number of $0$ digits in $b(\lambda)$.
This interpretation, with $N$ and $E$ in place of $0$ and $1$ respectively, is
used by Fu and Tang in [7] to prove theorems for partitions with fixed
perimeter.

Let $t_{n}(M)$ be the number of partitions of $n$ with perimeter $M$. Since
partitions having perimeter $M$ are in bijection with bit strings of length
$M+1$ where the first digit is $1$ and last digit is $0$ and there are
$2^{M-1}$ such bit strings, there are $2^{M-1}$ partitions with perimeter $M$.

######  Example 2.

$\displaystyle t_{5}(5)=$
$\displaystyle|\\{(5),(4,1),(3,1,1),(2,1,1,1),(1,1,1,1,1)\\}|=5,$
$\displaystyle t_{6}(5)=$ $\displaystyle|\\{(4,2),(3,2,1),(2,2,1,1)\\}|=3,$
$\displaystyle t_{7}(5)=$
$\displaystyle|\\{(4,3),(3,3,1),(3,2,2),(2,2,2,1)\\}|=4,$ $\displaystyle
t_{8}(5)=$ $\displaystyle|\\{(4,4),(3,3,2),(2,2,2,2)\\}|=3,$ $\displaystyle
t_{9}(5)=$ $\displaystyle|\\{(3,3,3)\\}|=1$

and $t_{n}(5)=0$ if $n\geq 10$. Thus, there are $16=2^{4}$ partitions with
perimeter $5$.

The generating function for $t_{n}(M)$ is given in the next theorem. It likely
appears in the literature but we were unable to find a mention of it.

######  Theorem 2.1.

We have

$\sum_{n,M\geq 0}t_{n}(M)\,z^{M}q^{n}=\sum_{M\geq
0}z^{M}q^{M}\sum_{j=0}^{M-1}\begin{bmatrix}M-1\\\ j\end{bmatrix}_{q}.$

###### Proof.

If $p(N,j,n)$ denotes the number of partitions of $n$ into at most $j$ parts,
each less than or equal to $N$, then from [1] we have

$\begin{bmatrix}M-1\\\
j\end{bmatrix}_{q}=\sum_{n=0}^{\infty}p(M-j-1,j,n)q^{n}.$

Given $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{\ell})$, a partition
of $n$ with $\ell\leq j$ parts, each less than or equal to $M-j-1$, let $\mu$
be the partition obtained from $\lambda$ by adding an outer hook of arm-length
$M-j$ and leg-length $j$, i.e.,
$\mu=(M-j,\lambda_{1}+1,\lambda_{2}+1,\ldots\lambda_{\ell}+1,1,\ldots,1$,
where the multiplicity of $1$ is $j-\ell$ (and it could be $0$). Then $\mu$
has perimeter $M$ and size $n+M$. ∎

We note that Staub’s bijection in his proof of Theorem 1.3 is defined
recursively. Fu and Tang [7] give a new interpretation of Straub’s bijection
that is no longer recursive. We describe their bijection
$\psi:\mathcal{H}(M)\to\mathcal{G}(M)$ in the language of bit strings. Note
that the bit string of a partition with distinct parts has no repeated $0$’s.
The bit string for a partition with odd parts has the property that every
initial substring ending in $0$ contains an odd number of digits equal to $1$.
So, given $\lambda\in\mathcal{H}(M)$, the bit string $b(\lambda)$ is
transformed (working from left to right) as follows:

The first $1$ remains unchanged.

A digit $1$ that is preceded by a $1$ changes to $0$.

A substring $01$ changes to $11$.

The last $0$ remains unchanged.

The resulting bit string $b(\mu)$ defines a partition $\mu\in\mathcal{G}(M)$.
Define $\psi(\lambda)=\mu$.

##  3\. Proof of Theorem 1.1

###### Analytic proof.

We have

$\displaystyle\sum_{n,u\geq 0}^{\infty}\alpha_{u}^{(r)}(n)z^{u}q^{n}$
$\displaystyle=\sum_{j=1}^{\infty}\frac{zq^{rj}}{1-zq^{rj}}\,\prod_{k=1}^{\infty}\frac{1-q^{rk}}{1-q^{k}}$
$\displaystyle=\sum_{j,h\geq
1}(zq^{rj})^{h}\,\prod_{k=1}^{\infty}\frac{1-q^{rk}}{1-q^{k}}$
$\displaystyle=\frac{1}{(q;q)_{\infty}}\sum_{h\geq
1}z^{h}q^{rh}\prod_{\begin{subarray}{c}k=1\\\ k\neq
h\end{subarray}}(1-q^{rk})$ $\displaystyle=\sum_{h\geq
1}\frac{z^{h}q^{rh}}{1-q^{h}}\prod_{\begin{subarray}{c}k=1\\\ k\neq
h\end{subarray}}(1+q^{r}+q^{2r}+\cdots+q^{(k-1)r})$
$\displaystyle=\sum_{n,u\geq 0}^{\infty}\beta_{u}^{(r)}(n)z^{u}q^{n}.$

∎

###### Combinatorial proof.

We employ the exponential notation and write $(a^{m})$ for the partition with
$m$ parts all equal to $a$. Given two partitions $\eta$ and $\xi$, the
partition $\eta\cup\xi$ is the partition obtained by rearranging the parts of
$\eta$ and $\xi$, respecting multiplicities. We construct a bijection

$\psi:\mathcal{O}_{u}(n;r,1)\to\mathcal{D}_{u}(n;r,1).$

Let $\lambda\in\mathcal{O}_{u}(n;r,1)$ and suppose $d=rj$, $j\geq 1$, is the
part divisible by $r$. Write $\lambda=\mu\cup((rj)^{u})$, where
$\mu\in\mathcal{O}(n;r,0)$. We define
$\psi(\lambda)=\varphi(\mu)\cup(u^{rj})\in\mathcal{D}_{u}(n;r,1)$ where
$\varphi$ is Glasher’s bijection. Note that
$\varphi(\mu)\in\mathcal{D}(n;r,0)$ could have parts equal to $u$. These occur
less than $r$ times.

To see that $\psi$ is invertible, let $\eta\in\mathcal{D}_{u}(n;r,1)$ and
suppose that part $u$ occurs $m_{u}\geq r$ times in $\eta$. Express
$m_{u}=qr+s$, for unique $q\geq 1$, $0\leq s<r$, and write
$\eta=\xi\cup(u^{rs})$. We have $\xi\in\mathcal{D}(n;r,0)$. Then
$\psi^{-1}(\eta)=\varphi^{-1}(\xi)\cup((rs)^{u})\in\mathcal{O}_{u}(n;r,1)$
with $\varphi^{-1}(\xi)\in\mathcal{O}(n;r,0)$. ∎

##  4\. An Analytic proof of Theorem 1.4

In this section, we prove Theorem 1.4 and show how it leads to an analytic
proof of Theorem 1.6.

###### A formula for $g(M)$.

The generating function for partitions with odd parts where the power of $q$
is the number partitioned, the power of $z$ is the perimeter and the power of
$t$ is the number of parts, takes the form

$\sum_{n\geq 0}\frac{tz^{2n+1}q^{2n+1}}{(tzq;q^{2})_{n+1}}=\sum_{n\geq
0}tz^{2n+1}q^{2n+1}\sum_{s\geq 0}\begin{bmatrix}n+s\\\
n\end{bmatrix}_{q^{2}}(tzq)^{s}.$

The coefficient of $z^{M}$ in the above sum is

$\sum_{n\geq 0}q^{M}t^{M-2n}\begin{bmatrix}{M-n-1}\\\ n\end{bmatrix}_{q^{2}}.$

To obtain a formula for $g(M)$, we must set $q=1$, differentiate with respect
to $t$ and then set $t=1$. The outcome is

(5) $g(M)=\sum_{0\leq 2n\leq M-1}(M-2n)\binom{M-n-1}{n}.$

Formula (5) allows an easy computation of the generating function for $g(M)$.
Namely,

$\displaystyle\sum_{M\geq 0}g(M)\,z^{M}$ $\displaystyle=\sum_{M\geq
0}z^{M}\sum_{0\leq 2n\leq M-1}(M-2n)\binom{M-n-1}{n}$
$\displaystyle=\sum_{M,n\geq 0}z^{M+2n+1}(M+1)\binom{M+n}{n}$
$\displaystyle=\sum_{M\geq 0}{z^{M+1}(M+1)}{(1-z^{2})^{-(M+1)}}=\sum_{M\geq
1}{z^{M}M}{(1-z^{2})^{-M}}$
$\displaystyle=\frac{\frac{z}{1-z^{2}}}{\left(1-\frac{z}{1-z^{2}}\right)^{2}}=\frac{z-z^{3}}{(1-z-z^{2})^{2}}.$

∎

###### A formula for $h(M)$.

We proceed exactly as was done for $g(M)$. The three variable generating
function is

$\sum_{n\geq
0}(-tzq;q)_{n}z^{n+1}q^{n+1}t=\sum_{n=0}^{\infty}\sum_{j=0}^{n}\begin{bmatrix}{n}\\\
j\end{bmatrix}_{q}q^{\binom{j+1}{2}}t^{j}z^{j}z^{n+1}q^{n+1}t.$

The coefficient of $z^{M}$ in the sum above is

$\sum_{0\leq 2j\leq M-1}t^{j+1}q^{M+\binom{j}{2}}\begin{bmatrix}{M-j-1}\\\
j\end{bmatrix}_{q}.$

To obtain $h(M)$, we must set $q=1$, differentiate with respect to $t$ and
then set $t=1$. Hence

(6) $h(M)=\sum_{0\leq 2j\leq M-1}(j+1)\binom{M-j-1}{j}.$

Formula (6) facilitates an easy computation of the generating function for
$h(M)$. That is,

$\displaystyle\sum_{M\geq 0}h(M)\,z^{M}$ $\displaystyle=\sum_{M\geq
0}z^{M}\sum_{0\leq 2j\leq M-1}(j+1)\binom{M-j-1}{j}$
$\displaystyle=\sum_{M,j\geq 0}z^{M+2j+1}(j+1)\binom{M+j}{j}=\sum_{j\geq
0}{z^{2j+1}(j+1)}{(1-z)^{-(j+1)}}$ $\displaystyle=\sum_{j\geq
1}{z^{2j-1}j}{(1-z)^{-j}}=z^{-1}\frac{\frac{z^{2}}{1-z}}{\left(1-\frac{z^{2}}{1-z}\right)^{2}}=\frac{z-z^{2}}{(1-z-z^{2})^{2}}.$

∎

###### A formula for $g_{1}(M)$.

Recall that $g_{1}(M)$ is the number of perimeter $M$ partitions with odd
parts and one exception, i.e., exactly one part is even and this part may be
repeated. In the bivariate generating function below, the power of $q$ is the
number partitioned and the power of $z$ is the perimeter. We write the
generating function as two sums. In the first, the largest part is even; in
the second, the largest part is odd:

$\sum_{k\geq 1}\frac{z^{2k}q^{2k}}{(zq;q^{2})_{k}(1-zq^{2k})}+\sum_{k\geq
0}\frac{z^{2k+1}q^{2k+1}}{(zq;q^{2})_{k+1}}\sum_{j=1}^{k}\frac{zq^{2j}}{1-zq^{2j}}.$

Setting $q=1$, we obtain the generating function for $g_{1}(M)$:

$\displaystyle\sum_{M\geq 0}g_{1}(M)\,z^{M}$ $\displaystyle=\sum_{k\geq
1}\frac{z^{2k}}{(1-z)^{k+1}}+\sum_{k\geq 0}\frac{z^{2k+2}k}{(1-z)^{k+2}}$
$\displaystyle=\sum_{k\geq
0}\left(\frac{z^{2k+2}}{(1-z)^{k+2}}+\frac{z^{2k+2}k}{(1-z)^{k+2}}\right)$
$\displaystyle=\sum_{k\geq 0}\sum_{j\geq 0}z^{2k+2+j}\binom{k+1+j}{j}(k+1).$

Therefore, we find that

$g_{1}(M)=\sum_{k\geq 0}(k+1)\binom{M-k-1}{k+1}.$

We may also simplify the generating function for $g_{1}(M)$ so that

$\displaystyle\sum_{M\geq 0}g_{1}(M)\,q^{M}$ $\displaystyle=\sum_{M\geq
1}\frac{z^{2M}}{(1-z)^{M+1}}+\sum_{M\geq 0}\frac{z^{2M+2}M}{(1-z)^{M+2}}$
$\displaystyle=\frac{z^{2}}{(1-z)(1-z-z^{2})}+\frac{z^{4}}{(1-z)^{3}\left(1-\frac{z^{2}}{1-z}\right)^{2}}=\frac{z^{2}}{(1-z-z^{2})^{2}}.$

∎

###### A formula for $h_{1}(M)$.

Recall that $h_{1}(M)$ is the number of perimeter $M$ partitions with distinct
parts and one exception, i.e., exactly one part is repeated. Again, when
setting up the bivariate generating function, we must take into account
whether or not (i) the largest part is repeated or (ii) the largest part is
not repeated:

$\sum_{M\geq 1}\frac{z^{M+1}q^{2M}}{1-zq^{M}}\,(-zq;q)_{M-1}+\sum_{M\geq
2}z^{M}q^{M}\sum_{j=1}^{M-1}\frac{(-zq;q)_{M-1}}{1+zq^{j}}\,\frac{z^{2}q^{2j}}{1-zq^{j}}.$

Therefore, setting $q=1$ in the sum above, we obtain

$\displaystyle\sum_{M\geq 0}h_{1}(M)\,q^{M}$ $\displaystyle=\sum_{M\geq
1}\frac{z^{M+1}(1+z)^{M-1}}{1-z}+\sum_{M\geq
2}\sum_{j=1}^{M-1}\frac{z^{M+2}(1+z)^{M-2}}{1-z}$ $\displaystyle=\sum_{M\geq
1}\frac{z^{M+1}(1+z)^{M-1}}{1-z}+\sum_{M\geq
2}\frac{(M-1)z^{M+2}(1+z)^{M-2}}{1-z}$ $\displaystyle=\sum_{M\geq
1}\frac{z^{M+1}(1+z)^{M-1}}{1-z}+\sum_{M\geq
1}\frac{Mz^{M+3}(1+z)^{M-1}}{1-z}$ $\displaystyle=\sum_{M\geq
1}\frac{z^{M+1}(1+z)^{M-1}(1+z^{2}M)}{1-z}$
$\displaystyle=\frac{z}{1-z^{2}}\sum_{M\geq 1}(z(1+z))^{M}(1+z^{2}M)$
$\displaystyle=\frac{z(z+z^{2})}{(1-z^{2})(1-z-z^{2})}+\frac{z^{3}}{1-z^{2}}\cdot\frac{z+z^{2}}{(1-z-z^{2})^{2}}$
$\displaystyle=\frac{z^{2}}{(1-z-z^{2})^{2}}$ $\displaystyle=\sum_{M\geq
0}g_{1}(M)\,z^{M}.$

Hence, we have found the generating function and proved that
$h_{1}(M)=g_{1}(M)$. ∎

######  Remark 2.

The four assertions in Theorem 1.6 now follow for the following reason. Each
of the sequences $g(M),h(M),g_{1}(M),h_{1}(M)$ satisfy the same fourth order
recurrence relation because each generating function has identical denominator
$(1-z-z^{2})^{2}$. In addition,

$\displaystyle\sum_{M\geq 0}F_{M}\,z^{M}$
$\displaystyle=\frac{z(1-z-z^{2})}{(1-z-z^{2})^{2}},$
$\displaystyle\sum_{M\geq 0}MF_{M}\,z^{M}$
$\displaystyle=\frac{z+z^{3}}{(1-z-z^{2})^{2}},$ $\displaystyle\sum_{M\geq
0}\sum_{j=0}^{M}F_{j}F_{M-j}\,z^{M}$
$\displaystyle=\frac{z^{2}}{(1-z-z^{2})^{2}}.$

Hence, all the sequences in the four claims of Theorem 1.6 fulfil the same
fourth order recurrence. Consequently, the proof of the four claims only
require checking that they are true for $M\leq 4$.

##  5\. A combinatorial proof of Theorem 1.6

In the current section, we supply combinatorial arguments for the four
recurrence relations listed in Theorem 1.6.

###### Proof.

We first prove the recurrence relation (1) for $g(M)$:

$g(M)=g(M-1)+g(M-2)+F_{M-1}.$

Recall that if $\lambda\in\mathcal{G}(M)$, then every initial substring of
$b(\lambda$ ending in $0$ contains an odd number of digits equal to $1$. The
set $\mathcal{G}(M)$ can be partitioned so that
$\mathcal{G}(M)=\mathcal{G}^{\prime}(M)\sqcup\mathcal{G}^{\prime\prime}(M)$,
where $\mathcal{G}^{\prime}(M)$ consists of partitions
$\lambda\in\mathcal{G}(M)$ such that the second digit of $b(\lambda)$ is $0$
and $\mathcal{G}^{\prime\prime}(M)$ consists of partitions
$\lambda\in\mathcal{G}(M)$ such that the second digit of $b(\lambda)$ is $1$.

Given $\lambda\in\mathcal{G}^{\prime}(M)$, we remove the second digit of
$b(\lambda)$, which is $0$, to obtain the bit string $b(\mu)$ of a partition
$\mu\in\mathcal{G}(M-1)$. Moreover, $\ell(\mu)=\ell(\lambda)-1$. This gives a
bijection between $\mathcal{G}^{\prime}(M)$ and $\mathcal{G}(M-1)$. Thus, the
total number of parts in all partitions in $\mathcal{G}^{\prime}(M)$ equals
$g(M-1)+|\mathcal{G}(M-1)|=g(M-1)+F_{M-1}$.

Given $\lambda\in\mathcal{G}^{\prime\prime}(M)$, since the second digit in
$b(\lambda)$ is $1$, the third digit must also be $1$. We remove the second
and third digits of $b(\lambda)$, which are both equal to $1$, to obtain the
bit string $b(\mu)$ of a partition $\mu\in\mathcal{G}(M-2)$. Moreover,
$\ell(\mu)=\ell(\lambda)$. As a result, we obtain a length-preserving
bijection between $\mathcal{G}^{\prime\prime}(M)$ and $\mathcal{G}(M-2)$. The
total number of parts in all partitions in $\mathcal{G}^{\prime\prime}(M)$
equals $g(M-2)$. So, equation (1) holds for all $M\geq 2$.

Next, we prove the recurrence relation (2) for $h(M)$:

$h(M)=h(M-1)+h(M-2)+F_{M-2}.$

Recall that if $\lambda\in\mathcal{H}(M)$, then $b(\lambda)$ has no repeated
$0$’s. The set $\mathcal{H}(M)$ can be partitioned into
$\mathcal{H}(M)=\mathcal{H}^{\prime}(M)\sqcup\mathcal{H}^{\prime\prime}(M)$,
where $\mathcal{H}^{\prime}(M)$ consists of partitions
$\lambda\in\mathcal{H}(M)$ such that the second digit of $b(\lambda)$ is $1$,
and $\mathcal{H}^{\prime\prime}(M)$ consists of partitions
$\lambda\in\mathcal{H}(M)$ such that the second digit of $b(\lambda)$ is $0$.

Given $\lambda\in\mathcal{H}^{\prime}(M)$, we remove the second digit of
$b(\lambda)$, which is $1$, to obtain the bit string $b(\mu)$ of a partition
$\mu\in\mathcal{H}(M-1)$. Moreover, $\ell(\mu)=\ell(\lambda)$. This gives a
length-preserving bijection between $\mathcal{H}^{\prime}(M)$ and
$\mathcal{H}(M-1)$. Thus, the total number of parts of all partitions in
$\mathcal{H}^{\prime}(M)$ equals $h(M-1)$.

Given $\lambda\in\mathcal{H}^{\prime\prime}(M)$, since the second digit is
$0$, the third digit must be $1$. We remove the second and third digits from
$b(\lambda)$ to obtain the bit string $b(\mu)$ of a partition
$\mu\in\mathcal{H}(M-2)$. Moreover, $\ell(\mu)=\ell(\lambda)-1$. This lends
itself to a bijection between $\mathcal{H}^{\prime\prime}(M)$ and
$\mathcal{H}(M-2)$. The total number of parts of all partitions in
$\mathcal{H}^{\prime\prime}(M)$ equals
$h(M-2)+|\mathcal{H}^{\prime\prime}(M)|=h(M-2)+|\mathcal{H}(M-2)|=h(M-2)+F_{M-2}$.
Therefore, equation (2) holds for all $M\geq 2$.

Next, we prove the recurrence relation (3) for $g_{1}(M)$:

$g_{1}(M)=g_{1}(M-1)+g_{1}(M-2)+F_{M-1}.$

Recall that $\mathcal{G}_{1}(M)$ is the set of partitions with perimeter $M$
and all parts odd except one even part (possibly repeated). Thus, if
$\lambda\in\mathcal{G}_{1}(M)$, then $b(\lambda)$ has a unique initial
substring ending in $0$ and containing an even number of $1$’s. For example,
if $b(\lambda)=1110100$, then $\lambda\in\mathcal{G}_{1}(6)$ and the unique
initial substring ending in $0$ with an even number of $1$’s is $11101$. If
$b(\lambda)=1110000$, then there is no such initial substring. Thus, $\lambda$
has no even part and $\lambda\not\in\mathcal{G}_{1}(6)$. If
$b(\lambda)=111010110$, then there are two such substrings, $111010$ and
$111010110$. Thus, $\lambda$ has two even parts and
$\lambda\not\in\mathcal{G}_{1}(8)$.

We partition $\mathcal{G}_{1}(M)$ into
$\mathcal{G}_{1}(M)=\mathcal{G}^{\prime}_{1}(M)\sqcup\mathcal{G}^{\prime\prime}_{1}(M)\sqcup\mathcal{G}^{\prime\prime\prime}_{1}(M)$,
where $\mathcal{G}^{\prime}_{1}(M)$ consists of partitions
$\lambda\in\mathcal{G}_{1}(M)$ such that the second digit of $b(\lambda)$ is
$0$, $\mathcal{G}^{\prime\prime}_{1}(M)$ consists of partitions
$\lambda\in\mathcal{G}_{1}(M)$ such that the second and third digits of
$b(\lambda)$ are both $1$’s, and $\mathcal{G}^{\prime\prime\prime}_{1}(M)$
consists of partitions $\lambda\in\mathcal{G}_{1}(M)$ such that the second
digit of $b(\lambda)$ is $1$ and third digit is $0$.

If $\lambda\in\mathcal{G}^{\prime}_{1}(M)$, removing the second digit from
$b(\lambda)$ we obtain the bit string $b(\mu)$ of a partition
$\mu\in\mathcal{G}_{1}(M-1)$. This gives a bijection between
$\mathcal{G}^{\prime}_{1}(M)$ and $\mathcal{G}_{1}(M-1)$. Similarly, the
transformation that removes the second and third digits from a bit string
corresponding to a partition in $\mathcal{G}^{\prime\prime}_{1}(M)$ gives a
bijection between $\mathcal{G}^{\prime\prime}_{1}(M)$ and
$\mathcal{G}_{1}(M-2)$. Next, given a partition
$\lambda\in\mathcal{G}^{\prime\prime\prime}_{1}(M)$, we denote by
$b_{0}(\lambda)$ the maximal substring of $0$’s starting with the third digit
of $b(\lambda)$. Suppose $b_{0}(\lambda)$ has length $k$. Then $1\leq k\leq
M-1$. If $k<M-1$ then removing the substring $b_{0}(\lambda)$ from
$b(\lambda)$ we obtain a bit string $b(\mu)$ for a partition
$\mu\in\mathcal{G}^{\prime\prime}(M-k)$, i.e., $b(\mu)\in\mathcal{G}(M-k)$
with third digit equal to $1$. As shown above,
$\mathcal{G}^{\prime\prime}(M-k)$ is in bijection with $\mathcal{G}(M-k-2)$.
There is exactly one partition $\lambda$ with $b_{0}(\lambda)$ of length $M-1$
and we can map this partition to the empty partition $\emptyset$. Thus,
$\mathcal{G}^{\prime\prime\prime}_{1}(M)$ is in bijection with
$\mathcal{G}(M-3)\sqcup\mathcal{G}(M-4)\sqcup\cdots\sqcup\mathcal{G}(1)\sqcup\\{\emptyset\\}$.
Since $|\mathcal{G}(N)|=F_{N}$ for all $N$, we have

$\displaystyle g_{1}(M)=$ $\displaystyle
g_{1}(M-1)+g_{1}(M-2)+F_{M-3}+F_{M-4}+\cdots+F_{1}+1$ $\displaystyle=$
$\displaystyle g_{1}(M-1)+g_{1}(M-2)+F_{M-1}.\qquad$

Finally, we prove the recurrence relation (4) for $h_{1}(M)$:

$h_{1}(M)=h_{1}(M-1)+h_{1}(M-2)+F_{M-1}.$

Recall that $\mathcal{H}_{1}(M)$ is the set of partitions with perimeter $M$
with one part repeated and all other parts distinct. If
$\lambda\in\mathcal{H}_{1}(M)$ then either

  * (i)

$b(\lambda)$ has a single substring $\bar{b}_{0}(\lambda)$ of length $2\leq
k\leq M-2$ with all digits $0$’s having a prefix and suffix $1$ in
$b(\lambda)$, or

  * (ii)

$b(\lambda)=1000\dots 0$, i.e. in $b(\lambda)$ all digits except the first are
equal to $0$.

We partition $\mathcal{H}_{1}(M)$ into
$\mathcal{H}_{1}(M)=\mathcal{H}^{\prime}_{1}(M)\sqcup\mathcal{H}^{\prime\prime}_{1}(M)\sqcup\mathcal{H}^{\prime\prime\prime}_{1}(M)$
where $\mathcal{H}^{\prime}_{1}(M)$ consists of partitions
$\lambda\in\mathcal{H}_{1}(M)$ such that the second digit of $b(\lambda)$ is
$1$, $\mathcal{H}^{\prime\prime}_{1}(M)$ consists of partitions
$\lambda\in\mathcal{H}_{1}(M)$ such that the second digit of $b(\lambda)$ is
$0$ and the third digit is $1$, and $\mathcal{H}^{\prime\prime\prime}_{1}(M)$
consists of partitions $\lambda\in\mathcal{H}_{1}(M)$ such that the second and
third digits of $b(\lambda)$ are both $0$.

If $\lambda\in\mathcal{H}^{\prime}_{1}(M)$, removing the second digit from
$b(\lambda)$ we obtain the bit string $b(\mu)$ of a partition
$\mu\in\mathcal{H}_{1}(M-1)$. This gives a bijection between
$\mathcal{H}^{\prime}_{1}(M)$ and $\mathcal{H}_{1}(M-1)$. Similarly, the
transformation that removes the second and third digits from a bit string
corresponding to a partition in $\mathcal{H}^{\prime\prime}_{1}(M)$ gives a
bijection between $\mathcal{H}^{\prime\prime}_{1}(M)$ and
$\mathcal{H}_{1}(M-2)$. Next, consider a partition
$\lambda\in\mathcal{H}^{\prime\prime\prime}_{1}(M)$. If $\lambda$ is in case
(i) above, removing the substring $\bar{b}_{0}(\lambda)$ from $b(\lambda)$ we
obtain a bit string $b(\mu)$ for a partition
$\mu\in\mathcal{H}^{\prime}(M-k)$, i.e., $b(\mu)\in\mathcal{H}(M-k)$ with
second digit equal to $1$. As shown above, $\mathcal{H}^{\prime}(M-k)$ is in
bijection with $\mathcal{H}(M-k-1)$. Lastly, there is exactly one partition
$\lambda$ in the case (ii) above, and we can map this partition to the empty
partition $\emptyset$. Thus, $\mathcal{H}^{\prime\prime}_{1}(M)$ is in
bijection with
$\mathcal{H}(M-3)\sqcup\mathcal{H}(M-4)\sqcup\cdots\sqcup\mathcal{H}(1)\sqcup\\{\emptyset\\}$.
Since $|\mathcal{H}(N)|=F_{N}$ for all $N$, we have

$\displaystyle h_{1}(M)=$ $\displaystyle
h_{1}(M-1)+h_{1}(M-2)+F_{M-3}+F_{M-4}+\cdots+F_{1}+1$ $\displaystyle=$
$\displaystyle h_{1}(M-1)+h_{1}(M-2)+F_{M-1}.$

This completes the proof of Theorem 1.6. ∎

######  Remark 3.

Theorem 1.6 implies Theorem 1.4. We explain this for the sequence $h(M)$. Let
$E$ be the forward shift operator given by $Ew(M)=w(M+1)$. In general, if a
function $w(M)$ is annihilated by the linear operator
$L:=a_{0}+a_{1}E+\cdots+a_{r}E^{r}$ then it is true that the ordinary
generating function for $w(M)$ is a rational function whose denominator is the
polynomial $a_{r}+\cdots+a_{1}x^{r-1}+a_{0}x^{r}$. The recurrence (2) can be
reformulated as $(E^{2}-E-1)h(M-2)=F_{M-2}$. On the other hand, the Fibonacci
sequence satisfies $(E^{2}-E-1)F_{M-2}=0$. It is now immediate

$(E^{2}-E-1)^{2}h(M)=(E^{2}-E-1)(E^{2}-E-1)h(M-2)=(E^{2}-E-1)F_{M-2}=0.$

In light of this, we may write

$\sum_{M\geq
1}h(M)x^{M}=\frac{b_{0}+b_{1}x+b_{2}x^{2}+b_{3}x^{3}}{(1-x-x^{2})^{2}}$

for constants $b_{0},b_{1},b_{2}$ and $b_{3}$, which in turn are determined
from initial conditions. Since $h(0)=0,h(1)=1,h(2)=1,h(3)=3$, one can easily
check that

$\sum_{M\geq 1}h(M)\,x^{M}=\frac{x-x^{2}}{(1-x-x^{2})^{2}},$

as stated in Theorem 1.4.

##  6\. Interpretation of $g(M)-h(M)$

Define the index of a partition $\lambda$ to be
$\ell(\lambda)-1-\lfloor\lambda_{1}/2\rfloor$, i.e.,

$\text{(\\# of parts of $\lambda$)}-1-\left\lfloor\frac{\text{largest part of
$\lambda$}}{2}\right\rfloor.$

######  Remark 4.

Note that for a partition $\lambda$ into odd parts, the index of $\lambda$ is
precisely the $M2$-rank of $\lambda$ as defined by Berkovich and Garvan [5].
The $M2$-rank of a partition $\lambda$ is the rank of the partition $\mu$
whose Ferrers diagram has the same shape as the $2$-modular diagram of
$\lambda$.

######  Theorem 6.1.

$g(M)-h(M)$ is the sum of indices of all the partitions into odd parts with
perimeter $M$.

######  Example 3.

$g(4)-h(4)=8-5=3$. The partitions in question are $3+1,3+3$ and $1+1+1+1$
whose indices are $0,0$ and $3$, respectively. Thus the sum of the indices is
indeed equal to $3$.

###### Proof.

The generating function for partitions with odd parts where the power of $q$
is the number partitioned, the power of $z$ is the perimeter and the power of
$t$ is the index, takes the form

$\sum_{n\geq 0}\frac{t^{-n}z^{2n+1}q^{2n+1}}{(tzq;q^{2})_{n+1}}=\sum_{n\geq
0}t^{-n}z^{2n+1}q^{2n+1}\sum_{s\geq 0}\begin{bmatrix}n+s\\\
n\end{bmatrix}_{q^{2}}(tzq)^{s}.$

The coefficient of $z^{M}$ in the above is

$\sum_{n\geq 0}q^{M}t^{M-3n-1}\begin{bmatrix}M-n-1\\\ n\end{bmatrix}_{q^{2}}.$

To obtain the sum of all the indices, we must set $q=1$, differentiate with
respect to $t$ and then set $t=1$. The result is the finite sum

$\displaystyle\sum_{n\geq 0}(M-3n-1)\binom{M-n-1}{n}$ $\displaystyle=$
$\displaystyle\sum_{n\geq 0}(M-2n)\binom{M-n-1}{n}-\sum_{n\geq
0}(n+1)\binom{M-n-1}{n}$ $\displaystyle=$ $\displaystyle g(M)-h(M).$

∎

######  Remark 5.

It follows from Theorem 6.1 and Corollary 1.8 that the sum of the indices of
all partitions of fixed perimeter with odd parts is non-negative.

##  7\. Final remarks and questions

For $r\geq 2$, let $\mathcal{G}_{r}(M)$ be the set of all perimeter $M$
partitions with parts not divisible by $r$ and set
$g_{r}(M)=|\mathcal{G}_{r}(M)|$. Let and $\mathcal{H}_{r}(M)$ be the set of
all perimeter $M$ partitions with parts occurring $<r$ times and set
$h_{r}(M)=|\mathcal{H}_{r}(M)|$. It is easily checked that, for $r\geq 3$, the
analogue of Glaisher’s identity for partitions with fixed perimeter fails.
Below, we find the generating functions for $h_{r}(M)$ and $g_{r}(M)$, as well
as the generating functions for the number of partitions with fixed perimeter
and parts congruent to $d$ modulo $r$.

######  Theorem 7.1.

For $r\geq 2$, we have a generating function for $h_{r}(M)$ given by

$\sum_{M\geq
1}h_{r}(M)\,z^{M}=\frac{z+z^{2}+\cdots+z^{r-1}}{1-z-z^{2}-\cdots-z^{r}}.$

###### Proof.

In the bivariate generating function below, the exponent of $z$ keeps track of
the perimeter and the exponent of $q$ keeps track of the size of the
partition. If $h_{r}(n,M)$ denotes the number of partitions of $n$ with
perimeter $M$ and having parts repeated less that $r$ times, then

$\displaystyle\sum_{n\geq 0}\sum_{M\geq 1}h_{r}(n,M)z^{M}q^{n}=$
$\displaystyle\sum_{n\geq
1}z^{n-1}(zq^{n}+z^{2}q^{2n}+\cdots+z^{r-1}q^{(r-1)n})\prod_{j=1}^{n-1}(1+zq^{j}+z^{2}q^{2j}+\cdots+z^{r-1}q^{(r-1)j}).$

Setting $q=1$ we obtain

$\displaystyle\sum_{M\geq 1}h_{r}(M)z^{M}$ $\displaystyle=\sum_{n\geq
1}z^{n-1}\cdot\frac{z-z^{r}}{1-z}\cdot\left(\frac{1-z^{r}}{1-z}\right)^{n-1}$
$\displaystyle=\frac{z-z^{r}}{1-z}\cdot\frac{1}{1-\frac{z(1-z^{r})}{1-z}}$
$\displaystyle=\frac{z-z^{r}}{1-2z+z^{r+1}}.$

Since $1-z$ divides both the numerator and the denominator, we cancel and find

$\sum_{M\geq
1}h_{r}(M)\,z^{M}=\frac{z+z^{2}+\cdots+z^{r-1}}{1-z-z^{2}-\cdots-z^{r}}.$

∎

######  Theorem 7.2.

For $r\geq 2$, we have a generating function for $g_{r}(M)$ given by

$\sum_{M\geq
1}g_{r}(M)\,z^{M}=\frac{z}{1-2z}\cdot\frac{(1-z)^{r-1}-z^{r-1}}{((1-z)^{r-1}-z^{r})}.$

###### Proof.

First, we keep track of the size of the partitions with exponent $q$ so that
there comes a generating function

$\sum_{n\geq
0}\sum_{d=1}^{r-1}z^{nr+d}q^{nr+d}\prod_{j=1}^{nr+d}\frac{1}{1-zq^{j}}\prod_{i=1}^{n}(1-zq^{ir}).$

Now, set $q=1$ to get

$\displaystyle\sum_{M\geq 1}g_{r}(M)\,z^{M}$ $\displaystyle=\sum_{n\geq
0}\sum_{d=1}^{r-1}\frac{z^{nr+d}}{(1-z)^{n(r-1)+d}}=\sum_{d=1}^{r-1}\left(\frac{z}{1-z}\right)^{d}\sum_{n\geq
0}\left(\frac{z^{r}}{(1-z)^{r-1}}\right)^{n}$
$\displaystyle=\frac{z}{1-z}\frac{1-\left(\frac{z}{1-z}\right)^{r-1}}{1-\frac{z}{1-z}}\frac{1}{1-\frac{z^{r}}{(1-z)^{r-1}}}.$

This simplifies to the desired formulation as depicted in the theorem. ∎

######  Theorem 7.3.

Let $r\geq 2$ and $1\leq d<r$. Denote by $\mathcal{G}_{r}^{(d)}(M)$ the set of
partitions with perimeter $M$ and having parts congruent to $d$ modulo $r$,
and let’s denote $g_{r}^{(d)}(M)=|\mathcal{G}_{r}^{(d)}(M)|$. Then, the
generating function for $g_{r}^{(d)}(M)$ is

$\sum_{M\geq 1}g_{r}^{(d)}(M)\,z^{M}=\frac{z^{d}}{1-z-z^{r}}.$

In particular, $g_{r}^{(d)}(M)=g_{r}^{(d-1)}(M-1)$ for each $1\leq d<r$.

###### Proof.

We mimic the proof of the generating function for $g(M)$ in Theorem 1.6. In
the bivariate generating function below, the exponent of $z$ keeps track of
the perimeter and the exponent of $q$ keeps track of the size of the
partition. Then, if $g_{r}^{(d)}(n,M)$ denotes the number of partitions of $n$
with perimeter $M$ having parts congruent to $d$ modulo $r$, we have

$\sum_{n\geq 0}\sum_{M\geq 1}g_{r}^{(d)}(n,M)z^{M}q^{n}=\sum_{n\geq
0}z^{nr+d}q^{nr+d}\sum_{s\geq 0}\begin{bmatrix}n+s\\\
n\end{bmatrix}_{q^{r}}(zq)^{s}.$

Setting $q=1$ we obtain

$\displaystyle\sum_{M\geq 1}g_{r}^{(d)}(M)z^{M}$
$\displaystyle=z^{d}\sum_{s\geq 0}z^{s}\sum_{n\geq
0}\binom{n+s}{n}(z^{r})^{n}=z^{d}\sum_{s\geq 0}\frac{z^{s}}{(1-z^{r})^{s+1}}$
$\displaystyle=\frac{z^{d}}{1-z^{r}}\cdot\frac{1}{1-\frac{z}{1-z^{r}}}=\frac{z^{d}}{1-z-z^{r}}.$

∎

######  Remark 6.

If $r=2$, from Theorems 7.1 and 7.3 we obtain $g_{2}^{(1)}(M)=h_{2}(M)$, which
is Theorem 1.3.

We conclude by noting that, for fixed $M\geq 1$ and $r\geq 2$, one should be
able to verify that $g_{r}(M)\leq h_{r}(M)$ using the generating functions
from Theorems 7.1 and 7.2; for instance, by comparing the largest real roots
of the denominator polynomials. However, this is likely tedious and not very
illuminating. Instead, it would be more appealing to give a positive answer to
the following question.

Question. Is there a combinatorial proof that $g_{r}(M)\leq h_{r}(M)$ for all
integers $M\geq 1$ and $r\geq 2$?

## References

  * [1] G. E. Andrews, The Theory of Partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original.
  * [2] G. E. Andrews, Euler’s partition identity and two problems of George Beck, Math. Student 86 (2017), no. 1-2, 115–119.
  * [3] C. Ballantine and A. Welch, Beck-type companion identities for Franklin’s identity via a modular refinement, Discrete Math. 344 (2021), no. 8, Paper No. 112480, 11 pp.
  * [4] The On-Line Encyclopedia of Integer Sequences, Sequence A090867, https://oeis.org.
  * [5] A. Berkovich and F. G. Garvan, Some observations on Dyson’s new symmetries of partitions, J. Combin. Theory Ser. A 100 (2002), no. 1, 61–93.
  * [6] S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1623–1635.
  * [7] S. Fu and D. Tang, Partitions with fixed largest hook length, Ramanujan J. 45 (2018), no. 2, 375–390.
  * [8] A. Straub, Core partitions into distinct parts and an analogue of Euler’s theorem, European J. Combin. 57 (2016), 40–49.
  * [9] Jane Y. X. Yang, Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem, European J. Combin. 76 (2019), 62–72.


# How to predict a new physics scale and its uncertainty with the Inverse
Amplitude Method

Alexandre Salas-Bernárdez111Speaker. The author thanks his collaborators on
this project F. J. Llanes-Estrada, J. Oller and J. Escudero-Pedrosa. This work
was funded by MINECO:FPA2016-75654-C2-1-P, MICINN: PID2019-108655GB-I00,
PID2019-106080GB- C21, UCM: EB15/21 (Spain).

###### Abstract

Effective Field Theories such as HEFT, organized as momentum expansions, are a
controllable approximation to strong dynamics only near threshold, as they
miss exact elastic unitarity, reducing their predictive power at a higher
scale if small separations from the Standard Model are found at the LHC or
elsewhere. Unitarized chiral perturbation theory extends their reach to the
saturation of unitarity but, generally with unknown systematics. Here I
present a brief "handbook" on how to put to use the Inverse Amplitude Method
to predict a resonance (associated with a new physics scale in Beyond the
Standard Model searches) and more especially on how to control its systematic
uncertainties.

##  1 Introduction

It is seeming more and more likely that the LHC or other current accelerators
will not find direct evidence of new particles. This is why a huge effort on
studying the traces that the new physics leaves on the visible particle
content is being made. This study relies naturally on Effective Field Theories
(EFTs). However, the naive or direct use of an EFT will likely be of no use
for predicting the behaviour of scattered particles above the energies
hitherto reached by accelerators. This is so because an EFT scattering
amplitude does not respect unitarity (an illustration of this fact is shown in
Fig. 1). Several unitarization methods have been devised to solve this
problem, such as the K-matrix method; the $N/D$ method; variations of the
Bethe-Salpeter equation or the Inverse Amplitude Method (IAM) [1]. The
difference between them is purely empirical, since, usually, their predictions
can only be tested if higher energies are reached. This is why it appears
useful to control the systematic uncertainties of each method ab initio, so
that their predictions are also given within an error-band.

Figure 1: Prediction of the $\rho$ resonance with the IAM using only threshold
$\pi\pi$-scattering data. The pure EFT amplitude (in blue) completely misses
the resonance and fails to reproduce the experimental data (in orange, taken
from [2]) just above threshold. The IAM predicts the position of the resonance
with an error of about 10%.

All things said, the most successful unitarization methods in the market also
respect the analyticity properties of amplitudes, this ensures causality in
the effective theory (the “old” K- Matrix method does not). Among these
methods, the IAM is salient due to its simple and algebraic form (the $N/D$
method is based on integral equations and large-$N_{C}$ methods are only
approximate for certain symmetry breaking patterns). This is why, in [3], we
have laid out the systematic uncertainties of the IAM based on a
dispersive/analytic approach.
In this Proceedings, I try to make a clear summary on how to put to use the
IAM to obtain a (resonant) new physics scale and its theoretical uncertainty
from LHC scattering data (f.e. the scattering of $W_{L}W_{L}$ in Higgs EFT).
This exercise is relevant to assess if, given precision data is obtained at
the LHC or elsewhere, whether the new physics scale may be at the reach of
future colliders (like FCC) or not. In Section 2 I will present an "IAM
Uncertainties Handbook", i.e. a quick list of the uncertainties of the method
and how to calculate them if wanted.

##  2 Inverse Amplitude Method’s Uncertainties Handbook

Amplitudes of two-to-two-body scattering for identical particles and definite
isospin $I$, $T_{I}(s,t,u)$ (expressed in terms of the usual Mandelstam’s
variables $s$, $t$ and $u$), admit a low-energy simplification: the partial-
wave projection

$T_{I}(s,t,u)=32\pi\sum_{J=0}^{\infty}(2J+1)t_{IJ}(s)P_{J}(\cos\theta_{s})\;.$
(1)

The partial waves, $t_{IJ}(s)$, are easily tractable in the context of EFTs
since they can be Taylor-expanded at low energies as
$t(s)=t_{0}(s)+t_{1}(s)+...$222Here and in the rest of the paper I omit spin
and isospin indices., here only the terms up to NLO
($\mathcal{O}(p^{4})=\mathcal{O}(s^{2})$) are made explicit. Unitarity of the
$S$ matrix forces the partial waves to follow the relation $\text{Im
}t(s)=\sigma(s)|t(s)|^{2}$ for physical $\text{Re }(s)>0$ (here
$\sigma(s)=\sqrt{1-\frac{s}{4m^{2}}}$ is the partial-wave two-body phase space
factor for particles of mass $m$ and for $s\geq 4m^{2}$). However the EFT
expansion satisfies this relation only order by order, f.e. $\text{Im
}t_{1}(s)=\sigma(s)|t_{0}(s)|^{2}$. On the other hand, if one expands the
reciprocal or inverse amplitude $1/t=1/t_{0}-t_{1}/t_{0}^{2}+...$, the NLO
approximation for $t(s)$,

$t_{\text{IAM}}(s)=\frac{t_{0}^{2}(s)}{t_{0}(s)-t_{1}(s)}$ (2)

will be found to satisfy unitarity exactly thanks to the simple relation
$\text{Im }1/t(s)=-\sigma(s)$.
In [3], the uncertainties of the IAM are thoroughly studied. The master
formula on which [3] leans on is based on a dispersive/analytic approach and
amounts to

$G(s)=G(0)+G^{\prime}(0)s+\frac{1}{2}G^{\prime\prime}(0)s^{2}+PC(G)+\frac{s^{3}}{\pi}LC[G]+\frac{s^{3}}{\pi}RC[G],$
(3)

where $G(s)\equiv t_{0}^{2}(s)/t(s)$ and $PC(G)$ are the pole contributions to
the dispersive integral coming from zeroes of $t(s)$333Here
$LC[f](s)=\int_{-\infty}^{0}dz\frac{\text{Im }f(z)}{z^{3}(z-s)}$ and
$RC[f](s)=\int_{4m^{2}}^{\infty}dz\frac{\text{Im }f(z)}{z^{3}(z-s)}$, where
$m$ is the mass of the scattered particles.. With this formula at hand it is
possible to derive the IAM amplitude in Eq. (2) by employing the following
approximations on Eq. (3):

  * •

Taking the NLO approximation of the subtraction constants
$G^{(i)}(0)\simeq(t_{0}-t_{1})^{(i)}(0)$.

  * •

Neglecting the pole contributions of $G(s)$, i.e. zeroes of $t(s)$. These
zeroes are of two types and both appear for real positive $s$: below threshold
they are denominated Adler zeroes and above threshold they are called
Castillejo-Dalitz-Dyson (CDD) zeroes.

  * •

Approximating at NLO the imaginary part of $G(s)$ over the Left Cut (LC),
$\text{Im }G(s)\simeq-\text{Im }t_{1}(s)$. (The left cut appears on the whole
negative real-$s$ axis due to the partial wave projection, giving a
discontinuity in the imaginary part of $t(s)$).

  * •

Neglecting inelastic channels on the whole Right Cut (RC)444Examples of these
inelastic channels in hadron physics would be $K\bar{K}$ or $2n$-$\pi$
intermediate states for $\pi\pi$ scattering. In HEFT, for $W_{L}W_{L}$
scattering the inelastic channels would be $2n$-$W_{L}$ or $n$-$h$.. Notice
the IAM is exact on the RC if inelastic channels are turned off.

The next subsection is devoted to explain how these sources of uncertainty can
be estimated.

###  2.1 Control of uncertainties

Each source of uncertainty in the method will affect the prediction for the
resonance’s mass and hence the new physics scale predicted by it (see in [4] a
comprehensive way of constructing HEFT partial wave amplitudes). The
mathematical procedure is to quantify the absolute value of

$\Delta(s)G(s)\equiv\Big{(}\frac{t(s)-t_{\text{IAM}(s)}}{t_{\text{IAM}}(s)}\Big{)}\frac{t_{0}^{2}(s)}{t(s)}\;.$
(4)

Since it is the above factor that modifies the IAM’s condition for a resonance
as

$t_{0}(s_{R})-t_{1}(s_{R})+i\sigma(s_{R})t_{0}^{2}(s_{R})=\Delta(s_{R})G(s_{R})\;,$
(5)

for real $s_{R}$ (the resonance’s mass squared). The steps to quantify its
magnitude are:

  1. 1.

To include the NNLO correction on the subtraction constants, one should use
Resonance Effective theory and lean on [5] (Eq. (A4) there). However these
constants are not yet known in HEFT (examples can be found in [6]). One should
estimate the size of the $\mathcal{O}(s^{3})$ terms at the resonance’s squared
mass $s_{R}$ and equate it to $|\Delta(s_{R})G(s_{R})|$.

  2. 2.

To assess the uncertainty due to neglecting the Adler zeroes simply compute
the ratio $m^{4}/s^{2}_{R}$. The appearance of CDD zeroes can be checked by
solving the simple algebraic equation $\text{Re }t_{1}(s_{C})+t_{0}(s_{C})=0$
for $s_{C}$ (zero’s position). If a solution is found, the IAM must fail, but
can be easily modified as
$t_{\text{IAM}}=\frac{t_{0}^{2}}{t_{0}-t_{1}}\to\frac{t_{0}^{2}}{t_{0}-t_{1}+\frac{s}{s-s_{c}}\text{Re}(t_{1})}$.
This substitution solves the issue.

  3. 3.

The easiest (though the less reliable and crude among the ones presented in
[3]) way of assessing the uncertainty on the left cut is simply assigning to
its uncertainty the LC integral of $t_{1}$ since it is presumably an upper
bound [3], $|\Delta(s_{R})G(s_{R})|=s^{3}_{R}|LC[t_{1}](s_{R})|/\pi$.

  4. 4.

Inelastic channels fall into two types. The two-body ones can be treated using
the matrix form of the IAM (see [2]), and be fully taken into account.
$n$-body intermediate states can be controlled by evaluating the ratio of the
$n$-body phase space factor against the two-body one,
$R(s)=\phi_{n}(s)/(v^{2n-4}\phi_{2}(s))$, at the maximum value where one
trusts the EFT unitarization (namely $4\pi v$ for HEFT, where $v=246$ GeV).
Then one equates the displacement of the pole to
$|\Delta(s_{R})G(s_{R})|=R((4\pi v)^{2})\cdot
s^{3}_{R}|RC[t_{1}](s_{R})|/\pi$.

Finally, to compute the displacement of the pole, one has to treat each
contribution to $|\Delta(s_{R})G(s_{R})|$ separately depending on their
scaling just as is presented in Appendix A of [3].

## References

  * [1] T. N. Truong, Phys. Rev. Lett. 61, 2526 (1988) doi:10.1103/PhysRevLett.61.2526
  * [2] A. Gomez Nicola and J. R. Pelaez, (2002) doi:10.1103/PhysRevD.65.054009
  * [3] A. Salas-Bernárdez et al., (2021) doi:10.21468/SciPostPhys.11.2.020
  * [4] D. Espriu, F. Mescia and I. Asiáin, [arXiv:2109.02673 [hep-ph]].
  * [5] Z. H. Guo and J. J. Sanz-Cillero, (2009) doi:10.1103/PhysRevD.79.096006
  * [6] A. Pich, I. Rosell and J. J. Sanz-Cillero, (2020) doi:10.1103/PhysRevD.102.035012