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Humps in Motzkin paths and standard Young tableaux in a $(2,1)$-hook

Xiaomei Chen

TL;DR

The paper refines Regev's results by counting humps and peaks in Motzkin paths at fixed height $k$ and by enumerating $SYT_k(2,1;n)$ with fixed first-two-part difference, establishing deep combinatorial links between these objects. It introduces bijections between humps and Motzkin prefixes, derives explicit closed-form expressions for $H_{n,k}$ and $P_{n,k}$, and develops a Riordan-array and generating-function perspective that connects these counts to $SYT_k(2,1;n)$. A new combinatorial proof of Regev-type identities is provided, along with recurrences such as $T_{n+1,k}+T_{n,k}=M_{n,k}$ and $H_{n,2k-1}$ relations, and generating functions like $T_k(x)=\frac{1}{1+x}(xM(x))^{k+1}$ are obtained. Collectively, these results deepen the link between Motzkin-path statistics and hook-shaped SYT enumeration, offering refined tools and potential generalizations for related combinatorial structures.

Abstract

We calculate the number of humps and peaks in Motzkin paths with a given height, and calculate the number of standard Young tableaux (SYTs) in a $(2,1)$-hook with the difference of the first two parts fixed, which refine Regev's results in 2009. We also give new combinatorial proofs of Regev's results, and reveal some new recurrence relations related to humps, free Motzkin paths and SYTs.

Humps in Motzkin paths and standard Young tableaux in a $(2,1)$-hook

TL;DR

The paper refines Regev's results by counting humps and peaks in Motzkin paths at fixed height and by enumerating with fixed first-two-part difference, establishing deep combinatorial links between these objects. It introduces bijections between humps and Motzkin prefixes, derives explicit closed-form expressions for and , and develops a Riordan-array and generating-function perspective that connects these counts to . A new combinatorial proof of Regev-type identities is provided, along with recurrences such as and relations, and generating functions like are obtained. Collectively, these results deepen the link between Motzkin-path statistics and hook-shaped SYT enumeration, offering refined tools and potential generalizations for related combinatorial structures.

Abstract

We calculate the number of humps and peaks in Motzkin paths with a given height, and calculate the number of standard Young tableaux (SYTs) in a -hook with the difference of the first two parts fixed, which refine Regev's results in 2009. We also give new combinatorial proofs of Regev's results, and reveal some new recurrence relations related to humps, free Motzkin paths and SYTs.

Paper Structure

This paper contains 3 sections, 7 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The number of humps and peaks with height $k$
  3. The number of SYTs in a $(2,1)$-hook

Key Result

Theorem 1.1

Let $H_{n,k}$ (resp. $P_{n,k}$) be the total number of humps (resp. peaks) with height $k$ in all Motzkin paths of order $n$. Then for $n\geq 2$ and $k\geq 1$, we have where $\chi$ is the characteristic function. Moreover, the number $j$ in the sum tracks up to the number of flat steps of the Motzkin paths.

Figures (3)

  • Figure 1: An example of hump and peak, where the hump $P_1$ is colored red, and the peak $P_2$ is colored blue.
  • Figure 2: An example of $\psi$ with $(M,P)\in \mathcal{H}_{9,2}$, where the hump $P$ is colored red.
  • Figure 3: An example of $\varPhi$ with $(M,P)\in \mathcal{H}_{9,2}$, where the hump $P$ is colored red.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • proof : Proof of Equation (\ref{['equ:3']})
  • proof : Proof of Equation (\ref{['equ:4']}) and (\ref{['equ:5']})
  • proof : Proof of Theorem \ref{['thm:22']}
  • Remark 2.2
  • Corollary 2.3
  • Lemma 3.1
  • ...and 8 more