# Combinatorics on bounded free Motzkin paths and its applications

Hyunsoo Cho  Hyunsoo Cho, Institute of Mathematical Sciences, Ewha Womans
University, Seoul, Republic of Korea<EMAIL_ADDRESS>,  JiSun Huh  JiSun
Huh, Department of Mathematics, Ajou University, Suwon, Republic of Korea
<EMAIL_ADDRESS>,  Hayan Nam  Hayan Nam, Department of Mathematics,
Duksung Women’s University, Seoul, Republic of Korea<EMAIL_ADDRESS>and
Jaebum Sohn  Jaebum Sohn, Department of Mathematics, Yonsei University, Seoul,
Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper, we construct a bijection from a set of bounded free Motzkin
paths to a set of bounded Motzkin prefixes that induces a bijection from a set
of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give
bijections between a set of bounded cornerless Motzkin paths and a set of
$t$-core partitions, and a set of bounded cornerless symmetric Motzkin paths
and a set of self-conjugate $t$-core partitions. As an application, we get
explicit formulas for the number of ordinary and self-conjugate $t$-core
partitions with a fixed number of corners.

##  1\. Introduction

The main result of this paper is finding a bijection between two sets of paths
in a bounded strip, which have been studied by several researchers (for
example, see [1, 5, 6, 7, 10, 13]).

A _Motzkin path_ of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays
weakly above the $x$-axis and consists of steps $u=(1,1)$, $d=(1,-1)$, and
$f=(1,0)$, called _up_ , _down_ , and _flat_ steps, respectively. A _free
Motzkin path_ of length $n$ is a path which starts at $(0,0)$ or $(0,1)$, ends
at $(n,0)$, and consists of $u$, $d$, and $f$. A Motzkin path with no
restrictions on the end point is called a Motzkin prefix. For a given path, a
_peak_ is a point preceded by an up step and followed by a down step and a
_valley_ is a point preceded by a down step and followed by an up step. We say
that a path is _cornerless_ if it has no peaks or valleys.

For non-negative integers $m,r$, and $k$, let $\mathcal{F}(m,r,k)$ be the set
of free Motzkin paths of length $m+r$ with $r$ flat steps that are contained
in the strip $-\lfloor\frac{k}{2}\rfloor\leq
y\leq\lfloor\frac{k+1}{2}\rfloor$. We denote $\mathcal{M}(m,r,k)$ the set of
Motzkin prefixes of length $m+r$ with $r$ flat steps that are contained in the
strip $0\leq y\leq k$. We define $L_{k}$ to be one of the boundaries of each
path depending on the value of $k$. More specifically, for
$P\in\mathcal{F}(m,r,k)$, denote $L_{k}$ by

$y=\begin{cases}\lfloor\frac{k+1}{2}\rfloor&\text{if $k$ is odd},\\\
-\lfloor\frac{k}{2}\rfloor&\text{if $k$ is even}.\end{cases}$

Let $\overline{\mathcal{F}}(m,r,k)$ (resp. $\overline{\mathcal{M}}(m,r,k)$) be
the set of paths in $\mathcal{F}(m,r,k)$ (resp. $\mathcal{M}(m,r,k)$) which
touch the line $L_{k}$ (resp. $y=k$) so that

$\mathcal{F}(m,r,k)=\bigcup_{i=0}^{k}\overline{\mathcal{F}}(m,r,i)\quad\text{and}\quad\mathcal{M}(m,r,k)=\bigcup_{i=0}^{k}\overline{\mathcal{M}}(m,r,i).$

Our main theorem states the following.

######  Theorem 1.1.

For given non-negative integers $m,r$, and $k$, there is a bijection between
the sets $\overline{\mathcal{F}}(m,r,k)$ and $\overline{\mathcal{M}}(m,r,k)$.

To prove Theorem 1.1, we construct a map $\phi_{m,k}$ and show that it is
bijective in Sections 2.1 and 2.2.

Using the adjacency matrices of path graphs, Cigler [5] showed that

$|A_{n,k}|=|B_{n,k}|=\sum_{j\in\mathbb{Z}}(-1)^{j}\binom{n}{\lfloor\frac{n+(k+2)j}{2}\rfloor}$

and expected the existence of a simple bijection between $A_{n,k}$ and
$B_{n,k}$, where $A_{n,k}$ is the set of paths of length $n$ which consist of
$u$ and $d$ only, start at $(0,0)$, end on height $0$ or $-1$, and are
contained in the strip $-\lfloor\frac{k+1}{2}\rfloor\leq
y\leq\lfloor\frac{k}{2}\rfloor$ of width $k$, and $B_{n,k}$ is the set of
paths of length $n$ which consist of $u$ and $d$ only, start at $(0,0)$ and
are contained in the strip $0\leq y\leq k$. Recently, Gu and Prodinger [10]
and Dershowitz [7] found bijections between $A_{n,k}$ and $B_{n,k}$
independently. We note that Theorem 1.1 with no flat step (equivalently,
$r=0$) gives a new bijection between $A_{n,k}$ and $B_{n,k}$ since
$\mathcal{F}(n,0,k)$ can be obtained from $A_{n,k}$ by mirroring left and
right and flipping along the $x$-axis, and $\mathcal{M}(n,0,k)=B_{n,k}$ as it
is. We should mention that the bijection $\phi_{m,k}$ is inspired by the
bijection due to Gu and Prodinger, but there is a property that $\phi_{m,k}$
holds whereas Gu and Prodinger’s does not. This property is described in
Section 2.3.

Let $\overline{\mathcal{F}}_{\rm c}(m,r,k)$ be the set of cornerless free
Motzkin paths in $\overline{\mathcal{F}}(m,r,k)$ that never start with a down
(resp. up) step for odd (resp. even) $m$ and $\overline{\mathcal{M}}_{\rm
c}(m,r,k)$ be the set of cornerless Motzkin prefixes in
$\overline{\mathcal{M}}(m,r,k)$ that end with a flat step. In Section 3.1, we
show that $\phi_{m,k}$ induces a bijection between
$\overline{\mathcal{F}}_{\rm c}(m,r,k)$ and $\overline{\mathcal{M}}_{\rm
c}(m,r,k)$.

In Section 3.2, we combinatorially interpret $t$-core partitions by cornerless
Motzkin paths. We describe a bijection between a set of cornerless Motzkin
paths and a set of $t$-core partitions. As an application of this bijection,
we count the number of $t$-core partitions with $m$ corners. In Section 3.3,
we also count the number of self-conjugate $t$-core partitions with $m$
corners by constructing bijections between any pair of the following sets: a
set of cornerless free Motzkin paths, a set of cornerless symmetric Motzkin
paths, and a set of self-conjugate $t$-core partitions.

##  2\. Bijection

In this section, we recursively define a map

$\phi_{m,k}:\bigcup_{r\geq
0}\overline{\mathcal{F}}(m,r,k)\rightarrow\bigcup_{r\geq
0}\overline{\mathcal{M}}(m,r,k),$

according to the values of $m$ and $k$, and then show that it is bijective.
For simplicity, we define some notations first. For a path $P=p_{1}p_{2}\dots
p_{n}$, where each $p_{i}$ denotes the $i$th step in $P$, let

$\overline{P}:=\overline{p}_{1}\overline{p}_{2}\dots\overline{p}_{n}\quad\text{and}\quad\overleftarrow{P}:=\overline{p}_{n}\overline{p}_{n-1}\dots\overline{p}_{1},$

where $\overline{u}:=d,~{}\overline{d}:=u$, and $\overline{f}:=f$.

###  2.1. Map $\phi_{m,k}$

Now we define the map. Let $P$ be a path in the set
$\overline{\mathcal{F}}(m,r,k)$ for some $r\geq 0$, and $\gamma\geq 0$ denote
the maximum number such that $f^{\gamma}$ is a suffix of $P$.

  1. Case 0.

If $k=0$ or $k=1$, then the map is defined as

$\phi_{m,k}(P):=\overleftarrow{P}.$

We show the bijection $\phi_{m,1}$ in Figure 1.

$f^{\beta}$$B$$f^{\gamma}$

$f^{\gamma}$$\overleftarrow{B}$$f^{\beta}$$\updownarrow$

(a) For odd $m$

$f^{\epsilon}$$u$$f^{\beta}$$B$$f^{\gamma}$

$f^{\gamma}$$\overleftarrow{B}$$f^{\beta}$$d$$f^{\epsilon}$$\updownarrow$

(b) For even $m$

Figure 1. The bijections $\phi_{m,1}$ in Case 0

Now assume $k>1$. A _special step_ of $P$ is the first step ending on the line
$L_{k}$. We write $P$ as

(1) $P=Af^{\alpha}sf^{\beta}Bf^{\gamma},$

where $s$ is the special step, $\alpha\geq 0$ (resp. $\beta\geq 0$) is the
maximum number of consecutive flat steps right before (resp. after) the step
$s$, $A$ denotes the prefix of $P$ before the subpath $f^{\alpha}s$, and $B$
denotes the subpath between the subpaths $sf^{\beta}$ and $f^{\gamma}$. Note
that $A$ and $B$ never end with a flat step (See Figure 2).

$A$$f^{\alpha}$$s$$f^{\beta}$$B$$f^{\gamma}$$\frac{k+1}{2}$$-\frac{k-1}{2}$$L_{k}$
(a) For odd $k$

$A$$f^{\alpha}$$s$$f^{\beta}$$B$$f^{\gamma}$$\frac{k}{2}$$-\frac{k}{2}$$L_{k}$
(b) For even $k$

Figure 2. The division of $P$ for $k>1$

Let the last vertex on the line $L_{k}$ (resp. $y=k$) be the _turning point_
of a path in $\overline{\mathcal{F}}(m,r,k)$ (resp.
$\overline{\mathcal{M}}(m,r,k)$). We call the first step after the turning
point starting from the $x$-axis and heading away from the line $L_{k}$ the
_break step_ , and denote it by $b$. If $P$ has the break step $b$, let
$\delta\geq 0$ be the maximum number of consecutive flat steps right before
the step $b$ and we write $B$ as $B_{1}f^{\delta}bB_{2}$.

  1. Case 1.

Let $m$ and $k$ have the same parity with $k>1$.

    1. i)

If there is no break step, then we write $P$ as (1) and define the map as

(2) $\phi_{m,k}(P):=\begin{cases}Q&\text{if $k$ is odd},\\\
\overline{Q}&\text{if $k$ is even},\end{cases}$

where

$Q:=f^{\gamma}\overline{B}f^{\alpha}sAf^{\beta}.$

Note that $\phi_{m,k}(P)$ ends on the line $y=k$.

    2. ii)

If there is the break step $b$, then $P$ can be written as

(3) $P=Af^{\alpha}sf^{\beta}B_{1}f^{\delta}bB_{2}f^{\gamma}.$

Note that $B_{1}$ is a subpath starting from the line $L_{k}$ and ending at
the $x$-axis with a down (resp. up) step, and $B_{2}$ is a subpath starting
from the line $y=(-1)^{k}$ and ending at the $x$-axis with a non-flat step for
odd (resp. even) $k$. Define

(4) $\phi_{m,k}(P):=\begin{cases}Q\overline{C}&\text{if $k$ is odd},\\\
\overline{Q}\overline{C}&\text{if $k$ is even},\end{cases}$

where

$Q:=f^{\gamma}\overline{B_{1}}f^{\alpha}sAf^{\beta}b\quad\text{and}\quad
C:=\begin{cases}\phi_{m^{\prime},k^{\prime}}(\overline{B_{2}}f^{\delta})&\text{if
$k$ is odd},\\\ \phi_{m^{\prime},k^{\prime}}(B_{2}f^{\delta})&\text{if $k$ is
even}.\end{cases}$

Note that $m^{\prime}$ is odd in this case. The bijection in Case 1 is
illustrated in Figure 3.

$A$$f^{\alpha}$$s$$f^{\beta}$$B$$f^{\gamma}$$\frac{k+1}{2}$$-\frac{k-1}{2}$$L_{k}$

$f^{\gamma}$$\overline{B}$$f^{\alpha}$$s$$A$$f^{\beta}$$k$$\frac{k+1}{2}$$\updownarrow$

(a) Case 1–i) for odd $m$ and $k$

$A$$f^{\alpha}$$s$$f^{\beta}$$B_{1}$$f^{\delta}$$b$$B_{2}$$f^{\gamma}$$\frac{k+1}{2}$$-\frac{k-1}{2}$$L_{k}$

$f^{\gamma}$$\overline{B_{1}}$$f^{\alpha}$$s$$A$$f^{\beta}$$b$$\overline{C}$$k$$\frac{k+1}{2}$$\updownarrow$

(b) Case 1–ii) for odd $m$ and $k$

$A$$f^{\alpha}$$s$$f^{\beta}$$B$$f^{\gamma}$$\frac{k}{2}$$-\frac{k}{2}$$L_{k}$

$f^{\gamma}$$B$$f^{\alpha}$$\overline{s}$$\overline{A}$$f^{\beta}$$k$$\frac{k}{2}$$\updownarrow$

(c) Case 1–i) for even $m$ and $k$

$A$$f^{\alpha}$$s$$f^{\beta}$$B_{1}$$f^{\delta}$$b$$B_{2}$$f^{\gamma}$$\frac{k}{2}$$-\frac{k}{2}$$L_{k}$

$f^{\gamma}$$B_{1}$$f^{\alpha}$$\overline{s}$$\overline{A}$$f^{\beta}$$\overline{b}$$\overline{C}$$k$$\frac{k}{2}$$\updownarrow$

(d) Case 1–ii) for even $m$ and $k$

Figure 3. The bijection $\phi_{m,k}$ in Case 1

  1. Case 2.

Let $m$ and $k$ have different parity with $k>1$. In this case we write $A$ as
$A_{1}aA_{2}$, where $a$ is the first up (resp. down) step starting from the
$x$-axis (resp. $y=1$) in $P$ for odd (resp. even) $k$. Here, $A_{1}$ and
$A_{2}$ can be empty. Note that if $A_{2}$ is non-empty, then it never ends
with a flat step. Similar to the map in Case 1–ii), we define the map as (4),
where $Q$ and $C$ are given as follows.

    1. i)

If there is no break step, then $P$ can be written as

(5) $P=A_{1}aA_{2}f^{\alpha}sf^{\beta}Bf^{\gamma},$

and we set

(6)
$Q:=f^{\gamma}\overline{B}f^{\alpha}sA_{2}f^{\beta}\overline{a}\quad\text{and}\quad
C:=\begin{cases}\phi_{m^{\prime},k^{\prime}}(A_{1})&\text{if $k$ is odd},\\\
\phi_{m^{\prime},k^{\prime}}(\overline{A_{1}})&\text{if $k$ is
even}.\end{cases}$

    2. ii)

If there is the break step $b$, then $P$ can be written as

(7) $P=A_{1}aA_{2}f^{\alpha}sf^{\beta}B_{1}f^{\delta}bB_{2}f^{\gamma},$

and we set

(8)
$Q:=f^{\gamma}\overline{B_{1}}f^{\alpha}sA_{2}f^{\beta}\overline{a}\quad\text{and}\quad
C:=\begin{cases}\phi_{m^{\prime},k^{\prime}}(A_{1}\overline{b}\overline{B_{2}}f^{\delta})&\text{if
$k$ is odd},\\\
\phi_{m^{\prime},k^{\prime}}(\overline{A_{1}}bB_{2}f^{\delta})&\text{if $k$ is
even}.\end{cases}$

Note that $m^{\prime}$ is even in this case. The bijection in Case 2–ii) is
illustrated in Figure 4. By regarding $B_{1}$ as $B$ and $f^{\delta}bB_{2}$ as
$\emptyset$ in this figure, we see the bijection in Case 2–i).

$A_{1}$$a$$A_{2}$$f^{\alpha}$$s$$f^{\beta}$$B_{1}$$f^{\delta}$$b$$B_{2}$$f^{\gamma}$$\frac{k+1}{2}$$-\frac{k-1}{2}$$L_{k}$

$f^{\gamma}$$\overline{B_{1}}$$f^{\alpha}$$s$$A_{2}$$f^{\beta}$$\overline{a}$$\overline{C}$$\frac{k+1}{2}$$k$$A_{1}$$\overline{b}$$\overline{B_{2}}$$f^{\delta}$$\updownarrow$

(a) For even $m$ and odd $k$

$A_{1}$$a$$A_{2}$$f^{\alpha}$$s$$f^{\beta}$$B_{1}$$f^{\delta}$$b$$B_{2}$$f^{\gamma}$$\frac{k}{2}$$-\frac{k}{2}$$L_{k}$

$f^{\gamma}$$B_{1}$$f^{\alpha}$$\overline{s}$$\overline{A_{2}}$$f^{\beta}$$a$$\overline{C}$$\frac{k}{2}$$k$$\overline{A_{1}}$$b$$B_{2}$$f^{\delta}$$\updownarrow$

(b) For odd $m$ and even $k$

Figure 4. The bijection $\phi_{m,k}$ in Case 2–ii)

######  Lemma 2.1.

For given non-negative integers $m$ and $k$, the map $\phi_{m,k}$ is well-
defined.

###### Proof.

Let $P\in\overline{\mathcal{F}}(m,r,k)$. In Case 0, it is clear that
$\phi_{m,k}(P)=\overleftarrow{P}\in\overline{\mathcal{M}}(m,r,k)$. Now
consider Case 1–i). In this case, for a path $P$ as in (1), we define
$\phi_{m,k}(P)$ as (2). If $k$ is odd (resp. even), then $A$ is a subpath of
$P$ that starts from the line $y=1$ (resp. $x$-axis), ends on the line
$y=(-1)^{k-1}\lfloor(k-1)/2\rfloor$, and is contained in the strip
$-\lfloor(k-1)/2\rfloor\leq y\leq\lfloor k/2\rfloor$, while $B$ is a subpath
that starts from the line $y=(-1)^{k-1}\lfloor(k+1)/2\rfloor$, ends on the
$x$-axis, and is contained in the strip $-\lfloor k/2\rfloor\leq
y\leq\lfloor(k+1)/2\rfloor$. Hence, the prefix
$f^{\gamma}\overline{B}f^{\alpha}$ (resp. $f^{\gamma}Bf^{\alpha}$) of
$\phi_{m,k}(P)$ is a Motzkin prefix that ends at the line
$y=\lfloor(k+1)/2\rfloor$ and is contained in the strip $0\leq y\leq k$. It
follows that the remaining path $sAf^{\beta}$ (resp. $\overline{sA}f^{\beta}$)
starts from the line $y=\lfloor(k+1)/2\rfloor$, ends on the line $y=k$, and is
contained in the strip $1\leq y\leq k$. Hence, the prefix
$f^{\gamma}\overline{B}f^{\alpha}$ (resp. $f^{\gamma}Bf^{\alpha}$) of
$\phi_{m,k}(P)$ is a Motzkin prefix that ends at the line
$y=\lfloor(k+1)/2\rfloor$ and is contained in the strip $0\leq y\leq k$, and
the remaining subpath $sAf^{\beta}$ (resp. $\overline{sA}f^{\beta}$) starts
from the line $y=\lfloor(k+1)/2\rfloor$, ends on the line $y=k$, and is
contained in the strip $1\leq y\leq k$ for odd (resp. even) $k$. Therefore,
$\phi_{m,k}(P)\in\overline{\mathcal{M}}(m,r,k)$.

For the remaining cases, we write $A$ as $A_{1}aA_{2}$ and $B$ as
$B_{1}f^{\delta}bB_{2}$ if necessary. Now we use the induction on $k$. For any
$k^{\prime}<k$, suppose that
$\phi_{m^{\prime},k^{\prime}}(P^{*})\in\overline{\mathcal{M}}(m^{\prime},r^{\prime},k^{\prime})$
for any paths
$P^{*}\in\overline{\mathcal{F}}(m^{\prime},r^{\prime},k^{\prime})$. Let
$P\in\overline{\mathcal{F}}(m,r,k)$, $\phi_{m,k}(P)$ is defined as
$Q\overline{C}$ or $\overline{Q}\overline{C}$, where $Q$ and $C$ are of the
forms in (4), (6), or (8). In any cases, similar to Case 1–i), $Q$ or
$\overline{Q}$ is a prefix of $\phi_{m,k}(P)$ that starts from the $x$-axis,
touches the line $y=k$, ends on the line $y=k-1$, and is contained in the
strip $0\leq y\leq k$. Since
$C\in\overline{\mathcal{M}}(m^{\prime},r^{\prime},k^{\prime})$ with
$k^{\prime}<k$, $\overline{C}$ is a suffix of $\phi_{m,k}(P)$ that starts from
the line $y=k-1$ and is contained in the strip $k-k^{\prime}-1\leq y\leq k-1$
by the induction hypothesis. Thus, we conclude that
$\phi_{m,k}(P)\in\overline{\mathcal{M}}(m,r,k)$. ∎

######  Example 2.2.

For given free Motzkin paths, let us apply the map $\phi_{m,k}$.

  1. (a)

For the path

$P_{1}=fduduufdfduufdff\in\overline{\mathcal{F}}(10,6,2),$

by applying Case 1–ii) and Case 0, we get

$\phi_{10,2}(P_{1})=ffuduufdfdufuddf\in\overline{\mathcal{M}}(10,6,2)$

since $A=\emptyset$, $\beta=\delta=0$, and $C=\phi_{1,1}(B_{2}f^{\delta})$ in
(3), and $\phi_{1,1}(fd)=\overleftarrow{fd}=uf$.

  2. (b)

For the path

$P_{2}=fduduufudfdduufudfdffdfufuddfuf\in\overline{\mathcal{F}}(20,11,3),$

by applying Case 2-ii), we obtain

$\phi_{20,3}(P_{2})=fufuuddfdufufudffduddfufudfduuf\in\overline{\mathcal{M}}(20,11,3)$

since $A_{2}=\emptyset$, $\beta=0$, and
$C=\phi_{10,2}(A_{1}\overline{b}\overline{B_{2}}f^{\delta})=\phi_{10,2}(P_{1})$
in (7), where $P_{1}$ is the path given in _(a)_.

See Figure 5 for further details.

$f^{\alpha}$$s$$B_{1}$$b$$B_{2}$$f^{\gamma}$$1$$-1$$L_{2}$

$\leftrightarrow$$f^{\gamma}$$B_{1}$$f^{\alpha}$$\overline{s}$$\overline{b}$$\overline{C}$$2$$0$

(a) A bijection $\phi_{10,2}$ in Case 1–ii)

$A_{1}$$a$$f^{\alpha}$$s$$B_{1}$$f^{\delta}$$b$$B_{2}$$f^{\gamma}$$2$$-1$$L_{3}$

$\updownarrow$$f^{\gamma}$$\overline{B_{1}}$$f^{\alpha}$$s$$\overline{a}$$\overline{C}$$3$$0$

(b) A bijection $\phi_{20,3}$ in Case 2–ii)

Figure 5. Examples of the map $\phi_{m,k}$

###  2.2. Map $\psi_{m,k}$

Now we define a map

$\psi_{m,k}:\bigcup_{r\geq
0}\overline{\mathcal{M}}(m,r,k)\rightarrow\bigcup_{r\geq
0}\overline{\mathcal{F}}(m,r,k)$

and show that $\psi_{m,k}=\phi_{m,k}^{-1}$. Let $S$ be a path in the set
$\overline{\mathcal{M}}(m,r,k)$ for some $r\geq 0$.

  1. Case 0.

For $k=0$ or $1$, we define $\psi_{m,k}(S)=\overleftarrow{S}$.

Recall that the last vertex on the line $y=k$ is called the turning point of a
path in $\overline{\mathcal{M}}(m,r,k)$. We define a _critical point_ of $S$
as the rightmost point on the $x$-axis which locates before the turning point.

  1. Case I.

For $k>1$, assume that $S$ is a path which ends on the line $y=k$. Note that
$m$ and $k$ have the same parity and we write

$S=f^{\gamma}B^{*}f^{\alpha}u^{*}A^{*}f^{\beta},$

where $u^{*}$ is the first up step starting from the line
$y=\lfloor(k+1)/2\rfloor$ after the critical point of $S$, $\gamma\geq 0$
(resp. $\beta\geq 0$) is the maximum number of consecutive initial (resp.
final) flat steps of $P$, and $\alpha\geq 0$ is the maximum number of
consecutive flat steps before the step $u^{*}$. Hence, $B^{*}$ (resp. $A^{*}$)
is the subpath of $S$ such that it starts from the $x$-axis (resp. $y=\lfloor
k/2\rfloor+1$), ends on the line $y=\lfloor k/2\rfloor$ (resp. $y=k$), and is
contained in the strip $0\leq y\leq k$ (resp. $1\leq y\leq k$). We define

(9)
$\psi_{m,k}(S):=\begin{cases}A^{*}f^{\alpha}u^{*}f^{\beta}\overline{B^{*}}f^{\gamma}&\text{if
$k$ is odd},\\\
\overline{A^{*}}f^{\alpha}\overline{u^{*}}f^{\beta}B^{*}f^{\gamma}&\text{if
$k$ is even}.\end{cases}$

  2. Case II.

Suppose that $S$ is a path which does not end on the line $y=k$ for $k>1$. In
this case, we write

$S=f^{\gamma}B^{*}f^{\alpha}u^{*}A^{*}f^{\beta}d^{*}\overline{C^{*}},$

where $u^{*},A^{*},B^{*},\alpha,\beta,\gamma$ is defined as in Case I, $d^{*}$
is the last down step starting from the line $y=k$, and $\overline{C^{*}}$ is
a suffix of $S$ after the step $d^{*}$. Note that
$C^{*}\in\overline{\mathcal{M}}(m^{\prime},r^{\prime},k^{\prime})$ for some
$k^{\prime}<k$ since $C^{*}$ is contained in the strip $0\leq y\leq k-1$.

    1. i)

Let $m$ and $k$ have the same parity, which follows that $m^{\prime}$ is odd.
We write $\psi_{m^{\prime},k^{\prime}}(C^{*})=B^{\bullet}f^{\delta}$, where
$\delta\geq 0$ is the maximum number of consecutive flat steps at the suffix
of $\psi_{m^{\prime},k^{\prime}}(C^{*})$. We set

(10)
$\psi_{m,k}(S):=\begin{cases}A^{*}f^{\alpha}u^{*}f^{\beta}\overline{B^{*}}f^{\delta}d^{*}\overline{B^{\bullet}}f^{\gamma}&\text{if
$k$ is odd},\\\
\overline{A^{*}}f^{\alpha}\overline{u^{*}}f^{\beta}B^{*}f^{\delta}\overline{d^{*}}B^{\bullet}f^{\gamma}&\text{if
$k$ is even}.\end{cases}$

    2. ii)

Let $m$ and $k$ have different parity. In this case, $m^{\prime}$ is even. We
divide two cases whether $\psi_{m^{\prime},k^{\prime}}(C^{*})$ goes above the
$x$-axis or not.

If $\psi_{m^{\prime},k^{\prime}}(C^{*})$ does not go above the $x$-axis, then
we write $\psi_{m^{\prime},k^{\prime}}(C^{*})=A^{\bullet}$ and define

(11)
$\psi_{m,k}(S):=\begin{cases}A^{\bullet}\overline{d^{*}}A^{*}f^{\alpha}u^{*}f^{\beta}\overline{B^{*}}f^{\gamma}&\text{if
$k$ is odd},\\\
\overline{A^{\bullet}}d^{*}\overline{A^{*}}f^{\alpha}\overline{u^{*}}f^{\beta}B^{*}f^{\gamma}&\text{if
$k$ is even}.\end{cases}$

If $\psi_{m^{\prime},k^{\prime}}(C^{*})$ goes above the $x$-axis, then we
write
$\psi_{m^{\prime},k^{\prime}}(C^{*})=A^{\bullet}u^{\bullet}B^{\bullet}f^{\delta}$,
where $u^{\bullet}$ is the first up step starting from the $x$-axis and
$\delta\geq 0$ is the maximum number of consecutive flat steps at the suffix
of $\psi_{m^{\prime},k^{\prime}}(C^{*})$. We define

(12)
$\psi_{m,k}(S):=\begin{cases}A^{\bullet}\overline{d^{*}}A^{*}f^{\alpha}u^{*}f^{\beta}\overline{B^{*}}f^{\delta}\overline{u^{\bullet}}\overline{B^{\bullet}}f^{\gamma}&\text{if
$k$ is odd},\\\
\overline{A^{\bullet}}d^{*}\overline{A^{*}}f^{\alpha}\overline{u^{*}}f^{\beta}B^{*}f^{\delta}u^{\bullet}B^{\bullet}f^{\gamma}&\text{if
$k$ is even}.\end{cases}$

######  Lemma 2.3.

The map $\psi_{m,k}$ is the inverse map of $\phi_{m,k}$.

###### Proof.

For $k=0$ or $1$, it is clear that $\psi_{m,k}(\phi_{m,k}(P))=P$ for any path
$P\in\overline{\mathcal{F}}(m,r,k)$ by the construction.

From now on, we set $k>1$. Let $P\in\overline{\mathcal{F}}(m,r,k)$ when $m$
and $k$ have the same parity and there is no break step in $P$ so that $P$ is
represented as $P=Af^{\alpha}sf^{\beta}Bf^{\gamma}$. When $k$ is odd (resp.
even), it follows from (2) and (9) that $\psi_{m,k}(\phi_{m,k}(P))=P$ since
$A=A^{*}$ (resp. $A=\overline{A^{*}}$), $s=u^{*}$ (resp.
$s=\overline{u^{*}}$), and $B=\overline{B^{*}}$ (resp. $B=B^{*}$), where
$S=\phi_{m,k}(P)\in\overline{\mathcal{M}}(m,r,k)$.

Now, we assume that
$\psi_{m^{\prime},k^{\prime}}(P^{*})\in\overline{\mathcal{F}}(m^{\prime},r^{\prime},k^{\prime})$
for any path $P^{*}$ with $k^{\prime}<k$. Let
$P\in\overline{\mathcal{F}}(m,r,k)$ when $m$ and $k$ have the same parity and
there is a break step $b$ in $P$ so that $P$ is represented as
$P=Af^{\alpha}sf^{\beta}B_{1}f^{\delta}bB_{2}f^{\gamma}$. For odd (resp. even)
$k$, according to (4) and (10), $\psi_{m,k}(\phi_{m,k}(P))=P$ since $A=A^{*}$
(resp. $A=\overline{A^{*}}$), $s=u^{*}$ (resp. $s=\overline{u^{*}}$),
$B_{1}=\overline{B^{*}}$ (resp. $B_{1}=B^{*}$), $b=d^{*}$ (resp.
$b=\overline{d^{*}}$), and $B_{2}=\overline{B^{\bullet}}$ (resp.
$B_{2}=B^{\bullet}$).

Similarly, by (6), (8), (11), and (12), we can see that
$\psi_{m,k}(\phi_{m,k}(P))=P$, where $P\in\overline{\mathcal{F}}(m,r,k)$ when
$m$ and $k$ have different parity with $k>1$. ∎

######  Example 2.4.

For given Motzkin prefixes,

$\displaystyle S_{1}$
$\displaystyle=uufufdddufuuf\in\overline{\mathcal{M}}(9,4,3),$ $\displaystyle
S_{2}$ $\displaystyle=uufuufdddfdfuuudfddf\in\overline{\mathcal{M}}(14,6,4),$
$\displaystyle S_{3}$
$\displaystyle=fuuufuduufuffdddfddfuuufufdddufuuf\in\overline{\mathcal{M}}(23,11,6),$

we have

$\displaystyle\psi_{9,3}(S_{1})$
$\displaystyle=ufddfdfuuudfd\in\overline{\mathcal{F}}(9,4,3),$
$\displaystyle\psi_{14,4}(S_{2})$
$\displaystyle=dfdfuuuufddfdfuuudfd\in\overline{\mathcal{F}}(14,6,4),$
$\displaystyle\psi_{23,6}(S_{3})$
$\displaystyle=ufufddduddfdfdffuuuuufddfdfuuudfdf\in\overline{\mathcal{F}}(23,11,6).$

See Figure 6 for further details.

$B^{*}$$u^{*}$$f^{\beta}$$3$$0$

$\updownarrow$$u^{*}$$f^{\beta}$$\overline{B^{*}}$$2$$-1$$L_{3}$

(a) A bijection $\psi_{9,3}$ in Case I

$B^{*}$$f^{\alpha}$$u^{*}$$A^{*}$$f^{\beta}$$d^{*}$$\overline{C^{*}}$$4$$0$

$\updownarrow$$\overline{A^{*}}$$f^{\alpha}$$\overline{u^{*}}$$f^{\beta}$$B^{*}$$\overline{d^{*}}$$B^{\bullet}$$2$$-2$$L_{4}$

(b) A bijection $\psi_{14,4}$ in Case II–i)

$f^{\gamma}$$B^{*}$$f^{\alpha}$$u*$$A^{*}$$f^{\beta}$$d^{*}$$\overline{C^{*}}$$6$$0$

$\updownarrow$$\overline{A^{\bullet}}$$d^{*}$$\overline{A^{*}}$$f^{\alpha}$$\overline{u^{*}}$$f^{\beta}$$B^{*}$$u^{\bullet}$$B^{\bullet}$$f^{\gamma}$$3$$-3$$L_{6}$

(c) A bijection $\psi_{23,6}$ in Case II–ii)

Figure 6. Examples of the map $\psi_{m,k}$

###  2.3. A property of $\phi_{m,k}$

For a free Motzkin path $P$, a maximal subpath in $P$ with no down (resp. up)
step is called an _upward_ (resp. _downward_) _run_ if it contains at least
one up (resp. down) step. Let $\operatorname{run}(P)$ denote the total number
of runs in $P$. If $P$ has no flat step, then the total number of peaks and
valleys of $P$ is counted by $\operatorname{run}(P)-1$. For example, the path
$P=uufufdddufuuf$ has two upward runs, $uufuf$ and $ufuuf$, and one downward
run $fddd$ so that $\operatorname{run}(P)=3$. Note that
$\operatorname{run}(P)=0$ if and only if $P$ is empty or a path consisting of
flat steps only, and $\operatorname{run}(P)=\operatorname{run}(\overline{P})$.
For a path $P\in\overline{\mathcal{F}}(m,r,k)$, the following proposition
shows that $\operatorname{run}(P)$ and $\operatorname{run}(\phi_{m,k}(P))$ are
differ by at most $1$.

######  Proposition 2.5.

For positive integers $m$ and $k$, let $P\in\overline{\mathcal{F}}(m,r,k)$ be
given.

  1. (a)

If $P$ starts with an upward run, then

$\operatorname{run}(\phi_{m,k}(P))=\operatorname{run}(P)-\\{1-(-1)^{m}\\}/2.$

  2. (b)

If $P$ starts with a downward run, then

$\operatorname{run}(\phi_{m,k}(P))=\operatorname{run}(P)-\\{1+(-1)^{m}\\}/2.$

###### Proof.

As erasing any number of flat steps do not change the number of runs, it
suffices to show that this proposition holds when $r=0$. We prove it by using
induction on $k$.

For the initial step with $k=1$, we consider Case 0. Recall that
$\phi_{m,1}(P)=\overleftarrow{P}$. If $P$ starts with an up step, $m$ must be
even and $\operatorname{run}(\overleftarrow{P})=\operatorname{run}(P)$. When
$P$ starts with a down step, $m$ is odd and
$\operatorname{run}(\overleftarrow{P})=\operatorname{run}(P)$.

Now we assume $k>1$ and suppose that this proposition holds for any
$P^{*}\in\overline{\mathcal{F}}(m^{\prime},0,k^{\prime})$ with $k^{\prime}<k$.
Here we give a detailed proof for (a) and the proof for (b) comes out
similarly.

Suppose that $P\in\overline{\mathcal{F}}(m,0,k)$ starts with an up step and
let $S=\phi_{m,k}(P)$. We need to show that $\operatorname{run}(S)$ is given
by $\operatorname{run}(P)-1$ (resp. $\operatorname{run}(P)$) if $m$ is odd
(resp. even).

In Case 1–i), we write $P=AuB$ (resp. $P=AdB$) and $S=\overline{B}uA$ (resp.
$S=Bu\overline{A}$), where $\overline{B}$ (resp. $B$) ends with an up step if
$m$ is odd (resp. even). If $A$ is empty, then $m$ must be odd so that
$\operatorname{run}(S)=\operatorname{run}(B)=\operatorname{run}(P)-1$ as we
desire. Now assume that $A$ starts with an up step. In this case,
$\operatorname{run}(P)=\operatorname{run}(A)+\operatorname{run}(B)$ and

$\operatorname{run}(S)=\begin{cases}\operatorname{run}(A)+\operatorname{run}(B)-1&\mbox{if
$m$ is odd},\\\ \operatorname{run}(A)+\operatorname{run}(B)&\mbox{if $m$ is
even},\end{cases}$

so we are done.

In Case 1–ii), we write $P=AuB_{1}dB_{2}$ (resp. $P=AdB_{1}uB_{2}$) and
$\overline{B_{2}}\in\overline{\mathcal{F}}(m^{\prime},k^{\prime})$ (resp.
$B_{2}\in\overline{\mathcal{F}}(m^{\prime},k^{\prime})$) for some
$k^{\prime}<k$ and odd $m^{\prime}$, where $m$ is odd (resp. even). Let
$r:=\operatorname{run}(A)+\operatorname{run}(B_{1})$. Note that if $m$ is odd
(resp. even), then

$\operatorname{run}(P)=\begin{cases}r+\operatorname{run}(B_{2})&\mbox{if
$B_{2}$ starts with an up (resp. down) step},\\\
r+\operatorname{run}(B_{2})-1&\mbox{if $B_{2}$ starts with a down (resp. up)
step}.\end{cases}$

In this case, if $m$ is odd (resp. even), then
$S=\overline{B_{1}}uAd\overline{C}$ (resp.
$S=B_{1}u\overline{A}d\overline{C}$), where
$C=\phi_{m^{\prime},k^{\prime}}(\overline{B_{2}})$ (resp.
$C=\phi_{m^{\prime},k^{\prime}}(B_{2})$). By the induction hypothesis, if $m$
is odd (resp. even), then

$\operatorname{run}(C)=\begin{cases}\operatorname{run}(B_{2})&\mbox{if $B_{2}$
starts with an up (resp. down) step},\\\ \operatorname{run}(B_{2})-1&\mbox{if
$B_{2}$ starts with a down (resp. up) step}.\\\ \end{cases}$

Hence, if $m$ is odd, then $\operatorname{run}(S)=r+\operatorname{run}(C)-1$
so that

$\operatorname{run}(S)=\begin{cases}r+\operatorname{run}(B_{2})-1&\mbox{if
$B_{2}$ starts with an up step},\\\ r+\operatorname{run}(B_{2})-2&\mbox{if
$B_{2}$ starts with a down step},\end{cases}$

which means that $\operatorname{run}(S)=\operatorname{run}(P)-1$. Similarly,
we show that $\operatorname{run}(S)=\operatorname{run}(P)$ for even $m$.

The proofs of Case 2–i) and Case 2–ii) are similar, so we only prove Case
2–ii). We divide this case into two cases depending on the parity of $m$.

When $m$ is odd, we write $P=A_{1}dA_{2}dB_{1}uB_{2}$ and
$S=B_{1}u\overline{A_{2}}d\overline{C}$, where $A_{1}$ starts with an up step
and $C=\phi_{m^{\prime},k^{\prime}}(\overline{A_{1}}uB_{2})$ for some
$k^{\prime}<k$ and even $m^{\prime}$. Note that
$\operatorname{run}(P)=\operatorname{run}(A_{1})+\operatorname{run}(dA_{2}dB_{1})+\operatorname{run}(uB_{2})-2$,
$\operatorname{run}(S)=\operatorname{run}(B_{1}u\overline{A_{2}}d)+\operatorname{run}(C)-1$,
and $\operatorname{run}(C)=\operatorname{run}(\overline{A_{1}}uB_{2})-1$ by
the induction hypothesis. We have

$\operatorname{run}(P)=\begin{cases}\operatorname{run}(A_{1})+\operatorname{run}(dA_{2}dB_{1})+\operatorname{run}(B_{2})-1&\mbox{if
$B_{2}$ starts with a down step},\\\
\operatorname{run}(A_{1})+\operatorname{run}(dA_{2}dB_{1})+\operatorname{run}(B_{2})-2&\mbox{if
$B_{2}$ starts with an up step},\end{cases}$

and

$\operatorname{run}(S)=\begin{cases}\operatorname{run}(B_{1}u\overline{A_{2}}d)+\operatorname{run}(A_{1})+\operatorname{run}(B_{2})-2&\mbox{if
$B_{2}$ starts with a down step},\\\
\operatorname{run}(B_{1}u\overline{A_{2}}d)+\operatorname{run}(A_{1})+\operatorname{run}(B_{2})-3&\mbox{if
$B_{2}$ starts with an up step}.\end{cases}$

Since
$\operatorname{run}(dA_{2}dB_{1})=\operatorname{run}(B_{1}u\overline{A_{2}}d)$
whenever $A_{2}$ starts with $u$ or $d$, we get
$\operatorname{run}(S)=\operatorname{run}(P)-1$.

For even $m$, we write $P=uA_{2}uB_{1}dB_{2}$ and
$S=\overline{B_{1}}uA_{2}d\overline{C}$, where
$C=\phi_{m^{\prime},k^{\prime}}(u\overline{B_{2}})$ for some $k^{\prime}<k$
and even $m^{\prime}$. We have $A_{1}=\emptyset$ because $P$ starts with $u$.
We also get $\operatorname{run}(S)=\operatorname{run}(P)$ in a similar manner.
∎

######  Remark 2.6.

Proposition 2.5 confirms that
$|\operatorname{run}(P)-\operatorname{run}(\phi_{m,k}(P))|\leq 1$, which shows
that the map $\phi_{m,k}$ is structurally distinguishable from the map from Gu
and Prodinger [10]. For example, Gu and Prodinger’s map sends

$P=dduuuuudddddduuu\mapsto S=uuuuuuddduuuddud,$

which shows that their map has paths satisfying
$|\operatorname{run}(P)-\operatorname{run}(S)|=2$ (one can obtain this example
by putting $A=dddddduuu$, $B=uu$, $C=du$, and $D=\emptyset$ in Figure 2.5 in
[10]).

Let $\overline{\mathcal{F}}(m,r,k;i)$ denote the set of paths in
$\overline{\mathcal{F}}(m,r,k)$ with $\lceil i/2\rceil$ downward (resp.
upward) runs for odd (resp. even) $m$, and let
$\overline{\mathcal{M}}(m,r,k;i)$ denote the set of paths in
$\overline{\mathcal{M}}(m,r,k)$ with $i$ runs. By Proposition 2.5, it is
straightforward to get the following corollary.

######  Corollary 2.7.

For non-negative integers $i$ and $m$ of the same (resp. different) parity,
the map $\phi_{m,k}$ induces a bijection between the set
$\overline{\mathcal{M}}(m,r,k;i)$ and the set of paths in
$\overline{\mathcal{F}}(m,r,k;i)$ that end with a downward (resp. upward) run.

######  Remark 2.8.

It is clear that the set of paths in $\overline{\mathcal{M}}(m,0,k;i)$ is in
bijection with the set of symmetric Dyck paths of length $2m$ with $i$ peaks
which touch the line $y=k$ and is contained in the strip $0\leq y\leq k$. By
Corollary 2.7, the set of symmetric Dyck paths of length $2m$ with $i$ peaks
corresponds to the set of paths in $\cup_{k}\overline{\mathcal{F}}(m,0,k;i)$
that end with a downward (resp. upward) run whenever $i$ and $m$ have the same
(resp. different) parity. This one-to-one correspondence gives a combinatorial
proof of the well-known fact that the number of symmetric Dyck paths of length
$2m$ with $i$ peaks is given by

(13)
$\binom{\lfloor\frac{m-1}{2}\rfloor}{\lfloor\frac{i-1}{2}\rfloor}\binom{\lfloor\frac{m}{2}\rfloor}{\lfloor\frac{i}{2}\rfloor}.$

##  3\. Cornerless free Motzkin paths

In this section, we combinatorially interpret cornerless Motzkin paths as a
$t$-core partitions. First let us consider the restriction of the map
$\phi_{m,k}$.

###  3.1. Restriction to cornerless free Motkzin paths

Recall that $\overline{\mathcal{F}}_{\rm c}(m,r,k)$ is the set of cornerless
free Motzkin paths in $\overline{\mathcal{F}}(m,r,k)$ that never start with a
down (resp. up) step for odd (resp. even) $m$ and $\overline{\mathcal{M}}_{\rm
c}(m,r,k)$ is the set of cornerless Motzkin prefixes in
$\overline{\mathcal{M}}(m,r,k)$ that end with a flat step. Now we show that
the map $\phi_{m,k}$, defined in Section 2.1, gives a one-to-one
correspondence between these sets.

######  Proposition 3.1.

For given non-negative integers $m,r$, and $k$, $\phi_{m,k}$ induces a
bijection between the sets $\overline{\mathcal{F}}_{\rm c}(m,r,k)$ and
$\overline{\mathcal{M}}_{\rm c}(m,r,k)$.

###### Proof.

Let $P\in\overline{\mathcal{F}}(m,r,k)$ and
$\phi_{m,k}(P)\in\overline{\mathcal{M}}(m,r,k)$. For $k=0$ or $1$,
$P\in\overline{\mathcal{F}}_{\rm c}(m,r,k)$ when it is cornerless and starts
with a flat step, and $\overleftarrow{P}\in\overline{\mathcal{M}}_{\rm
c}(m,r,k)$ when it is cornerless and ends with a flat step. Hence,
$P\in\overline{\mathcal{F}}_{\rm c}(m,r,k)$ if and only if
$\phi_{m,k}(P)=\overleftarrow{P}\in\overline{\mathcal{M}}_{\rm c}(m,r,k)$.

Let $m$ and $k$ have the same parity and there is no break step in $P$ with
$k>1$. It follows from (1) and (2) that $P\in\overline{\mathcal{F}}_{\rm
c}(m,r,k)$ and $\phi_{m,k}(P)\in\overline{\mathcal{M}}_{\rm c}(m,r,k)$ have
the same restriction such that $A$ and $B$ are cornerless, $A$ does not start
with a down (resp. up) step for odd (resp. even) $m$, and $\beta>0$. Hence,
$P\in\overline{\mathcal{F}}_{\rm c}(m,r,k)$ if and only if
$\phi_{m,k}(P)\in\overline{\mathcal{M}}_{\rm c}(m,r,k)$.

For the remaining cases, we assume that $\phi_{m^{\prime},k^{\prime}}$ induces
a bijection between $\overline{\mathcal{F}}_{\rm
c}(m^{\prime},r^{\prime},k^{\prime})$ and $\overline{\mathcal{M}}_{\rm
c}(m^{\prime},r^{\prime},k^{\prime})$ for $k^{\prime}<k$. We consider the case
when $m$ and $k$ have the same parity and there is a break step $b$ in $P$. By
(3) and (4), $P\in\overline{\mathcal{F}}_{\rm c}(m,r,k)$ and
$\phi_{m,k}(P)\in\overline{\mathcal{M}}_{\rm c}(m,r,k)$ have the same
condition such that $A$ and $B_{1}$ are cornerless, $A$ does not start with a
down (resp. up) step, $\overline{B_{2}}f^{\delta}$ (resp. $B_{2}f^{\delta}$)
$\in\overline{\mathcal{F}}_{\rm c}(m^{\prime},r^{\prime},k^{\prime})$ for some
$k^{\prime}<k$ when $m$ is odd (resp. even), and $\beta>0$. Hence,
$P\in\overline{\mathcal{F}}_{\rm c}(m,r,k)$ if and only if
$\phi_{m,k}(P)\in\overline{\mathcal{M}}_{\rm c}(m,r,k)$.

Similarly, we can show that $P\in\overline{\mathcal{F}}_{\rm c}(m,r,k)$ if and
only if $\phi_{m,k}(P)\in\overline{\mathcal{M}}_{\rm c}(m,r,k)$ when $m$ and
$k$ have different parity by considering (5), (6), (7), and (8). ∎

###  3.2. Cornerless Motkzin paths and $t$-cores

A _partition_ $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{\ell})$ is a
non-increasing positive integer sequence. The _Young diagram_ of $\lambda$ is
an array of boxes arranged in left-justified rows with $\lambda_{i}$ boxes in
the $i$th row. An _inner corner_ of a Young diagram is a box that can be
removed from the Young diagram and the rest of the Young diagram is still the
Young diagram of a partition. We say that $\lambda$ has $m$ corners if its
Young diagram has $m$ inner corners. For a given Young diagram, the _hook
length_ of a box at the position $(i,j)$, denoted by $h(i,j)$, is the number
of boxes on the right, in the below, and itself. For a partition $\lambda$,
the _beta-set_ of $\lambda$, denoted by $\beta(\lambda)$, is the set of hook
lengths of boxes in the first column of the Young diagram of $\lambda$. A
partition is called a _$t$ -core_ if its Young diagram has no box of hook
length $t$. We mainly consider $t$-core partitions with $m$ corners and use
the abacus diagram introduced by James and Kerber [12] to count them. The _$t$
-abacus diagram_ is a diagram to be the bottom and left-justified diagram with
infinitely many rows labeled by $i\in\mathbb{N}\cup\\{0\\}$ and $t$ columns
labeled by $j=0,1,\dots,t-1$ whose position $(i,j)$ is labeled by $ti+j$. The
_$t$ -abacus_ of a partition $\lambda$ is obtained from the $t$-abacus diagram
by placing a bead on each position labeled by $h$, where $h\in\beta(\lambda)$.
A position without bead is called a _spacer_. The following lemma is useful to
determine whether a given partition is $t$-core or not.

######  Lemma 3.2.

[12, Lemma 2.7.13] A partition $\lambda$ is a $t$-core if and only if
$h\in\beta(\lambda)$ implies $h-t\in\beta(\lambda)$ whenever $h>t$.
Equivalently, $\lambda$ is a $t$-core if and only if the $t$-abacus of
$\lambda$ has no spacer below a bead in any column.

From the above lemma, we easily obtain a simple bijection between the set of
$t$-core partitions and the set of non-negative integer sequences
$(n_{0},n_{1},\dots,n_{t-1})$, where $n_{0}=0$ and $n_{j}$ is the number of
beads in column $j$ for $j=1,\dots,t-1$. Using the bijection between the bar
graphs and cornerless Motzkin paths, introduced by Deutsch and Elizalde [8],
we give a path interpretation of the $t$-core partitions restricted by the
number of corners and the first hook length $h(1,1)$.

######  Theorem 3.3.

For non-negative integers $t$, $m$, and $k$, there is a bijection between any
pair of the following sets.

  1. (a)

The set of $t$-core partitions with $m$ corners such that $h(1,1)<kt$.

  2. (b)

The set of non-negative integer sequences $(n_{0},n_{1},\dots,n_{t-1})$
satisfying that $n_{0}=0$, $n_{i}\leq k$ for all $i$, and

$\sum_{i=1}^{t}|n_{i}-n_{i-1}|=2m,$

where we set $n_{t}:=0$.

  3. (c)

The set of cornerless Motzkin paths of length $2m+t-1$ with $t-1$ flat steps
that are contained in the strip $0\leq y\leq k$.

###### Proof.

Let $A,B$, and $C$ be the set described in (a), (b), and (c), respectively.
Set the maps $\phi_{1}:A\rightarrow B$ and $\phi_{2}:B\rightarrow C$. For a
partition $\lambda\in A$, let $n_{i}$ be the number of beads in the $i$th
column of the $t$-abacus of $\lambda$. Given $\lambda\in A$, define
$\phi_{1}(\lambda)=(n_{0},n_{1},\ldots,n_{t-1})$. Then, by the definition of
the $t$-abacus and the fact that $h(1,1)<kt$, it is given that $n_{0}=0$ and
$n_{i}\leq k$ for each $i$. Moreover, we get one inner corner for each maximal
sequence of consecutive numbers in the beta-set $\beta(\lambda)$. Note that
$\sum_{i=1}^{t}\max(n_{i}-n_{i-1},0)$ counts the number of hook lengths which
is the smallest among each maximal sequence of consecutive numbers in the
beta-set, so we get $\sum_{i=1}^{t}|n_{i}-n_{i-1}|=2m$. Let
$\psi_{1}:B\rightarrow A$ and $\vv{n}=(n_{0},n_{1},\ldots,n_{t-1})\in B$.
Define $\psi_{1}(\vv{n})=\lambda$, where $\lambda$ is the partition obtained
from the $t$-abacus diagram with $n_{i}$ beads in the $i$th column. We place
the beads on the elements of $\beta(\lambda)$ in the $t$-abacus diagram. Then,
since column $0$ has no bead and each $n_{i}\leq k$ for all $i$, the largest
element in $\beta(\lambda)$ is less than $kt$, meaning that $\lambda$ is a
$t$-core partition with $h(1,1)<kt$. Also, the fact that the sum of
$|n_{i}-n_{i-1}|$ is $2m$ implies that there are $m$ piles of beads which are
placed on $m$ maximal consecutive numbers, so $\lambda$ has $m$ corners.

For $\vv{n}=(n_{0},n_{1},\ldots,n_{t-1})\in B$, let
$\phi_{2}(\vv{n})=P_{\vv{n}}$, where $P_{\vv{n}}$ be the cornerless Motzkin
path which starts at $(0,0)$, ends at $(2m+t-1,0)$, and has $t-1$ flat steps
at height $n_{1},n_{2},\cdots,n_{t-1}$ with proper up and/or down steps
connecting those flat steps. Due to the fact that $n_{i}\leq k$, it is given
that $P_{\vv{n}}$ is contained in the strip $0\leq y\leq k$.

Let $\psi_{2}:C\rightarrow B$ and $P\in C$. Define
$\psi_{2}(P)=(n_{0},n_{1},\ldots,n_{t-1})$, where $n_{0}=0$ and, for $1\leq
i\leq t-1$, each $n_{i}$ represents the height of the $i$th flat step in $P$.
We know that $P$ is contained in the strip $0\leq y\leq k$, which implies
$n_{i}\leq k$. On the path $P$, there are $2m$ many up and down steps. The
number $|n_{i}-n_{i-1}|$ represents the difference of the height of the
$(i-1)$st flat step and the $i$th flat step, so it counts the number of up or
down steps in between those two flat steps. Since
$\sum_{i=1}^{t}\max(n_{i}-n_{i-1},0)=\sum_{i=1}^{t}|\min(n_{i}-n_{i-1},0)|$,
we get $\sum_{i=1}^{t}|n_{i}-n_{i-1}|=2m$.

∎

For example, there are sixteen $4$-core partitions with $2$ corners. By
letting $t=4$ and $m=2$ in Theorem 3.3, we get the correspondence between
these partitions, abaci, non-negative integer sequences, and cornerless
Motzkin paths as described in Figure 7.

$\ast$
(4,4,2,2)$\leftrightarrow$01234567$\leftrightarrow$[0,0,2,2]$\leftrightarrow$

(2,2,2,1,1,1)$\leftrightarrow$01234567$\leftrightarrow$[0,2,2,2]$\leftrightarrow$$\star$

(3,3,1,1,1)$\leftrightarrow$01234567$\leftrightarrow$[0,1,2,2]$\leftrightarrow$

(5,2,2)$\leftrightarrow$01234567$\leftrightarrow$[0,0,1,2]$\leftrightarrow$

(6,3)$\leftrightarrow$01234567$\leftrightarrow$[0,0,0,2]$\leftrightarrow$

(3,3,1,1)$\leftrightarrow$01234567$\leftrightarrow$[0,2,2,0]$\leftrightarrow$

(4,2,2)$\leftrightarrow$01234567$\leftrightarrow$[0,0,2,1]$\leftrightarrow$

(2,2,1,1,1)$\leftrightarrow$01234567$\leftrightarrow$[0,2,2,1]$\leftrightarrow$

$\ast$
(4,1,1,1)$\leftrightarrow$01234567$\leftrightarrow$[0,1,1,2]$\leftrightarrow$

(5,2)$\leftrightarrow$01234567$\leftrightarrow$[0,0,2,0]$\leftrightarrow$$\star$

(3,1,1,1)$\leftrightarrow$01234567$\leftrightarrow$[0,1,2,1]$\leftrightarrow$$\star$

(4,1,1)$\leftrightarrow$01234567$\leftrightarrow$[0,1,2,0]$\leftrightarrow$

(2,1,1,1)$\leftrightarrow$01234567$\leftrightarrow$[0,2,1,1]$\leftrightarrow$

$\ast$
(3,1,1)$\leftrightarrow$01234567$\leftrightarrow$[0,2,1,0]$\leftrightarrow$

(4,1)$\leftrightarrow$01234567$\leftrightarrow$[0,2,0,0]$\leftrightarrow$

$\ast$
(2,1)$\leftrightarrow$01234567$\leftrightarrow$[0,1,0,1]$\leftrightarrow$$\star$

Figure 7. $4$-cores with $2$ corners and the corresponding objects

We denote that a partition $\lambda$ is a $(t_{1},t_{2},\ldots,t_{p})$-core if
$\lambda$ is a $t_{i}$-core for all $i=1,\dots,p$. It is known that the number
of $t$-core partitions is infinite, and the number of
$(t_{1},t_{2},\ldots,t_{p})$-cores is finite for relatively prime
$t_{1},\dots,t_{p}$. Huang and Wang [11] enumerated the number of
$(t,t+1)$-cores, $(t,t+1,t+2)$-cores with the fixed number of corners, where
these results are generalized to $(t,t+1,\cdots,t+p)$-cores in [4]. As far as
we know, it seems new to get the formula for the number of $t$-core partitions
with the fixed number of corners, which we enumerate this by using the path
interpretation.

######  Proposition 3.4.

The number of $t$-core partitions with $m$ corners is given by

$\operatorname{cc}(t,m):=\sum_{i=1}^{\min(m,\lfloor
t/2\rfloor)}N(m,i)\binom{t+2m-2i}{2m},$

where $N(m,i)=\frac{1}{m}\binom{m}{i}\binom{m}{i-1}$ denotes the Narayana
number.

###### Proof.

By Theorem 3.3, $\operatorname{cc}(t,m)$ is equal to the number of cornerless
Motzkin paths of length $2m+t-1$ with $t-1$ flat steps. Let a Dyck path
consisting of $m$ up steps and $m$ down steps with $i$ peaks be given. The
number of ways of inserting $t-1$ flat steps such that the resultant path
becomes a cornerless Motzkin path is $\binom{t+2m-2i}{2m}$ since we have to
insert at least one flat steps at the positions of $i$ peaks and $i-1$
valleys. As the number of Dyck paths consisting of $m$ up steps and $m$ down
steps with $i$ peaks is counted by the Narayana number $N(m,i)$, the proof is
followed. ∎

The numbers of $t$-core partitions with $m$ corners for $2\leq t\leq 6$ and
$1\leq m\leq 8$ are given in Table 1. Clearly, $\operatorname{cc}(2,m)=1$,
$\operatorname{cc}(3,m)=2m+1$, and $\operatorname{cc}(4,m)=(5m^{2}+5m+2)/2$.
See sequences A063490 and A160747 in [14] for more the values of
$\operatorname{cc}(t,m)$ for $t=5$ and $t=6$, respectively.

$t\backslash m$ |  | 1 |  | 2 |  | 3 |  | 4 |  | 5 |  | 6 |  | 7 |  | 8
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
2 |  | 1 |  | 1 |  | 1 |  | 1 |  | 1 |  | 1 |  | 1 |  | 1
3 |  | 3 |  | 5 |  | 7 |  | 9 |  | 11 |  | 13 |  | 15 |  | 17
4 |  | 6 |  | 16 |  | 31 |  | 51 |  | 76 |  | 106 |  | 141 |  | 181
5 |  | 10 |  | 40 |  | 105 |  | 219 |  | 396 |  | 650 |  | 995 |  | 1445
6 |  | 15 |  | 85 |  | 295 |  | 771 |  | 1681 |  | 3235 |  | 5685 |  | 9325
Table 1. The numbers $\operatorname{cc}(t,m)$ of $t$-cores with $m$ corners

###  3.3. Cornerless symmetric Motzkin paths and self-conjugate $t$-cores

For a partition $\lambda$, its _conjugate_ is the partition
$\lambda^{\prime}=(\lambda^{\prime}_{1},\lambda^{\prime}_{2},\dots)$, where
each $\lambda^{\prime}_{j}$ is the number of boxes in the $j$th column of the
Young diagram of $\lambda$. A partition $\lambda$ is called _self-conjugate_
if $\lambda=\lambda^{\prime}$. Let $\operatorname{MD}(\lambda)$ denote the set
of the main diagonal hook lengths of $\lambda$. Note that if $\lambda$ is a
self-conjugate partition, then the elements in $\operatorname{MD}(\lambda)$
are all distinct and odd. Similar to Lemma 3.2, Ford, Mai, and Sze [9] gave a
useful result to determine whether a given partition is self-conjugate
$t$-core or not.

######  Proposition 3.5.

[9, Proposition 3] Let $\lambda$ be a self-conjugate partition. Then $\lambda$
is a $t$-core if and only if both of the following hold:

  1. (a)

For $h>t$, if $h\in\operatorname{MD}(\lambda)$, then
$h-2t\in\operatorname{MD}(\lambda)$.

  2. (b)

If $h_{1},h_{2}\in\operatorname{MD}(\lambda)$, then $h_{1}+h_{2}\neq 2t$.

We slightly modify the $t$-abacus to get the _$t$ -doubled abacus_, which is
useful when we deal with a self-conjugate $t$-core partition. Let the _$t$
-doubled abacus diagram_ is a left-justified diagram with infinitely many rows
labeled by $i\in\mathbb{Z}$ and $\lfloor t/2\rfloor$ columns labeled by
$j=0,1,\dots,\lfloor t/2\rfloor-1$ whose position $(i,j)$ is labeled by
$|2(ti+j)+1|$. The _$t$ -doubled abacus_ of a self-conjugate partition
$\lambda$ is obtained from the $t$-doubled abacus diagram by placing a bead on
each position labeled by $h$, where $h\in\operatorname{MD}(\lambda)$. From
Proposition 3.5, we have the following lemma.

######  Lemma 3.6.

A self-conjugate partition $\lambda$ is $t$-core if and only if the
$t$-doubled abacus diagram of $\lambda$ satisfies both of the following.

  1. (a)

If a bead is placed on position $(i,j)$ with $i>0$ (resp. $i<0$), then a bead
is also placed on position $(0,j)$ (resp. $(-1,j)$) and there is no spacer
between them in any column $j$.

  2. (b)

A bead can be placed on at most one of the two positions $(-1,j)$ and $(0,j)$
in any column $j$.

From the above lemma, we easily obtain a simple bijection between the set of
self-conjugate $t$-core partitions and the set of integer sequences
$(n_{0},\dots,n_{\lfloor t/2\rfloor-1})$, where the number of beads in column
$j$ is denoted by either $n_{j}$ or $-n_{j}$ for $j=0,1,\dots,\lfloor
t/2\rfloor-1$ if a bead is placed in position $(0,j)$ or not, respectively.
Now we give a path interpretation of the self-conjugate $t$-core partitions
restricted by the number of corners and the first hook length $h(1,1)$. We
define

$\mathcal{F}_{c}(m,r,k):=\bigcup_{i=0}^{k}\overline{\mathcal{F}}_{c}(m,r,i)\quad\text{and}\quad\mathcal{M}_{c}(m,r,k):=\bigcup_{i=0}^{k}\overline{\mathcal{M}}_{c}(m,r,i).$

######  Theorem 3.7.

For non-negative integers $t$, $m$, and $k$, there is a bijection between any
pair of the following sets.

  1. (a)

The set of self-conjugate $t$-cores with $m$ corners such that $h(1,1)<kt$.

  2. (b)

The set of integer sequences $(n_{0},n_{1},\dots,n_{\lfloor t/2\rfloor-1})$
satisfying that for odd (resp. even) $m$, $n_{0}$ is positive (resp. non-
positive); for all $i$, $-\lfloor k/2\rfloor\leq
n_{i}\leq\lfloor(k+1)/2\rfloor$; and

$\sum_{i=0}^{\lfloor t/2\rfloor}|n_{i}-n_{i-1}|=\begin{cases}m+1&\text{for
odd~{}}m,\\\ m&\text{for even~{}}m,\end{cases}$

where we set $n_{-1}:=0$ and $n_{\lfloor t/2\rfloor}:=0$.

  3. (c)

The set of cornerless free Motzkin paths in $\mathcal{F}_{\rm c}(m,\lfloor
t/2\rfloor,k)$.

  4. (d)

The set of cornerless Motzkin prefixes in $\mathcal{M}_{\rm c}(m,\lfloor
t/2\rfloor,k)$.

  5. (e)

The set of cornerless symmetric Motzkin paths of length $2m+t-1$ with $t-1$
flat steps that are contained in the strip $0\leq y\leq k$.

###### Proof.

Let $A,B,C,D,$ and $E$ be the set described in (a), (b), (c), (d), and (e),
respectively. By similar argument to the proof of Proposition 3.1, we know
that there is a bijection between $C$ and $D$. Now we set
$\phi_{1}:A\rightarrow B,\phi_{2}:B\rightarrow C$, and $\phi_{3}:D\rightarrow
E$ and show that $\phi_{1},\phi_{2},\phi_{3}$ are bijections.

Given $\lambda\in A$, let $\phi_{1}(A)=(n_{0},n_{1},\ldots,n_{\lfloor
t/2\rfloor-1})$, where each $n_{i}$ is the highest or lowest row that the bead
is placed in the $i$th column depending on the sign of $n_{i}$. We get that
$1\in MD(\lambda)$ when the number of corners $m$ is odd and $1\not\in
MD(\lambda)$ otherwise. Thus, $n_{0}$ is positive when $m$ is odd and negative
otherwise. This map gives a bijection between $A$ and $B$.

Let $\vv{n}=(n_{0},n_{1},\ldots,n_{\lfloor t/2\rfloor-1})$. For odd (resp.
even) $m$, let $\phi_{2}(\vv{n})$ be the cornerless free Motzkin path that
starts at $(0,1)$ (resp. $(0,0)$), ends at $(m+\lfloor t/2\rfloor,0)$, has
$i$th flat step at height $n_{i-1}$ with proper up and down steps between
them. Then, the map $\phi_{2}$ describes a bijection between $B$ and $C$.

Denote a path by $P=p_{1}p_{2}\cdots p_{m+\lfloor t/2\rfloor}\in D$. We set

$\phi_{3}(P)=\begin{cases}p_{1}p_{2}\cdots p_{m+\lfloor
t/2\rfloor}p_{m+\lfloor t/2\rfloor}\cdots p_{2}p_{1}&\text{if $m$ is odd,}\\\
p_{1}p_{2}\cdots p_{m+\lfloor t/2\rfloor-1}p_{m+\lfloor
t/2\rfloor}p_{m+\lfloor t/2\rfloor-1}\cdots p_{2}p_{1}&\text{if $m$ is
even.}\end{cases}$

Then, the map $\phi_{3}$ is a bijective. ∎

Note that Figure 7 shows that there are four self-conjugate $4$-core
partitions with $2$ corners and four cornerless symmetric Motzkin paths of
length $7$ with $3$ flat steps, which are marked by $\ast$ and $\star$,
respectively. The correspondences between the sets described in Theorem 3.7
for $t=4,m=2$ and $t=5,m=3$ are given in Figure 8.

(4,4,2,2)$\leftrightarrow$1375$\leftrightarrow$[-1,-1]$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$

(4,1,1,1)$\leftrightarrow$1375$\leftrightarrow$[-1,0]$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$

(3,1,1)$\leftrightarrow$1375$\leftrightarrow$[0,-1]$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$

(2,1)$\leftrightarrow$1375$\leftrightarrow$[0,1]$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$

(a) $t=4$ and $m=2$

(7,7,4,4,2,2,2)$\leftrightarrow$11131397$\leftrightarrow$[2,2]$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$

(7,3,3,1,1,1,1)$\leftrightarrow$11131397$\leftrightarrow$[1,2]$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$

(6,3,3,1,1,1)$\leftrightarrow$11131397$\leftrightarrow$[2,1]$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$

(6,2,1,1,1,1)$\leftrightarrow$11131397$\leftrightarrow$[2,0]$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$

(4,2,1,1)$\leftrightarrow$11131397$\leftrightarrow$[1,-1]$\leftrightarrow$$\leftrightarrow$$\leftrightarrow$

(b) $t=5$ and $m=3$

Figure 8. Examples of self-conjugate $t$-cores with $m$ corners and the
corresponding objects

Although the number of self-conjugate $(t,t+1,\cdots,t+p)$-cores with the
fixed number of corners is unknown in general, it is enumerated in [2, 3] when
$p=1,2,$ and $3$. The number of self-conjugate $t$-core partitions with $m$
corners can be counted by using these path interpretations.

######  Proposition 3.8.

The number of self-conjugate $t$-core partitions with $m$ corners is given by

$\operatorname{scc}(t,m):=\sum_{i=1}^{\min(m,\lfloor
t/2\rfloor)}\binom{\lfloor\frac{m-1}{2}\rfloor}{\lfloor\frac{i-1}{2}\rfloor}\binom{\lfloor\frac{m}{2}\rfloor}{\lfloor\frac{i}{2}\rfloor}\binom{\lfloor\frac{t}{2}\rfloor+m-i}{m}$

for $m>0$ and $\operatorname{scc}(t,0)=1$. In addition,
$\operatorname{scc}(t,m)=\operatorname{scc}(t+1,m)$ for even $t$.

###### Proof.

By Theorem 3.7, $\operatorname{scc}(t,m)$ also counts the number of cornerless
symmetric Motzkin paths of length $2m+t-1$ with $t-1$ flat steps. Let a
symmetric Dyck path consisting of $m$ up steps and $m$ down steps with $i$
peaks with $2i\leq t$ be given. The number of ways inserting $t-1$ flat steps
such that the resultant path becomes a cornerless symmetric Motzkin path is
$\binom{\lfloor t/2\rfloor+m-i}{m}$. The proof is followed since the number of
symmetric Dyck paths consisting of $m$ up steps and $m$ down steps with $i$
peaks is given by (13). ∎

The numbers of self-conjugate $t$-core partitions with $m$ corners for $2\leq
t\leq 11$ and $1\leq m\leq 8$ are given in Table 2. Clearly,
$\operatorname{scc}(2,m)=\operatorname{scc}(3,m)=1$,
$\operatorname{scc}(4,m)=\operatorname{scc}(5,m)=\lfloor 3m/2\rfloor+1$, and
$\operatorname{scc}(6,m)=\operatorname{scc}(7,m)=(10m(m+1)+(-1)^{m}(2m+1)+7)/8$.

$t\backslash m$ |  | 1 |  | 2 |  | 3 |  | 4 |  | 5 |  | 6 |  | 7 |  | 8
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
2,3 |  | 1 |  | 1 |  | 1 |  | 1 |  | 1 |  | 1 |  | 1 |  | 1
4,5 |  | 2 |  | 4 |  | 5 |  | 7 |  | 8 |  | 10 |  | 11 |  | 13
6,7 |  | 3 |  | 9 |  | 15 |  | 27 |  | 37 |  | 55 |  | 69 |  | 93
8,9 |  | 4 |  | 16 |  | 34 |  | 76 |  | 124 |  | 216 |  | 309 |  | 471
10,11 |  | 5 |  | 25 |  | 65 |  | 175 |  | 335 |  | 675 |  | 1095 |  | 1875
Table 2. The numbers $\operatorname{scc}(t,m)$ of self-conjugate $t$-cores
with $m$ corners

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