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

TL;DR

This paper investigates combinatorial properties of Motzkin paths and standard Young tableaux in a specific hook shape, providing new formulas, proofs, and recurrence relations that extend and refine previous results.

Contribution

It introduces new formulas and recurrence relations for counting humps, peaks, and SYTs in a (2,1)-hook, along with novel combinatorial proofs.

Findings

Derived formulas for humps and peaks in Motzkin paths with fixed height.

Counted SYTs in a (2,1)-hook with fixed first two parts difference.

Established new recurrence relations related to humps, Motzkin paths, and SYTs.

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.