On the enumeration of three-rowed standard Young tableaux of skew shape in terms of Motzkin numbers

TL;DR

This paper derives explicit generating functions for counting three-rowed skew standard Young tableaux, expressing these counts as linear combinations of Motzkin numbers and revealing connections to Chebyshev polynomials.

Contribution

It provides a general explicit formula for the generating functions of skew SYTs with at most three rows, extending previous conjectures and results.

Findings

Generated formulas relate to Motzkin numbers and Chebyshev polynomials.

Extended Regev's conjecture to a broader class of skew shapes.

Established connections between combinatorial enumeration and classical orthogonal polynomials.

Abstract

The enumeration of standard Young tableaux (SYTs) of shape {\lambda} can be easily computed by the hook-length formula. In 1981, Amitai Regev proved that the number of SYTs having at most three rows with n entries equals the nth Motzkin number M_n. In 2006, Regev conjectured that the total number of SYTs of skew shape {\lambda}/(2, 1) over all partitions {\lambda} having at most three parts with n entries is the difference of two Motzkin numbers, M_{n-1} - M_{n-3}. Ekhad and Zeilberger proved Regev's conjecture using a computer program. In 2009, S.-P. Eu found a bijection between Motzkin paths and SYTs of skew shape with at most three rows to prove Regev's conjecture, and Eu also indirectly showed that for the fixed {\mu} = ({\mu}1,{\mu}2) the number of SYTs of skew shape {\lambda}/{\mu} over all partitions {\lambda} having at most three parts can be expressed as a linear combination of the Motzkin numbers. In this paper, we will find an explicit formula for the generating function for the general case: for each partition {\mu} having at most three parts the generating function gives a formula for the coefficients of the linear combination of Motzkin numbers. We will also show that these generating functions are unexpectedly related to the Chebyshev polynomials of the second kind.