Combinatorial properties of the numbers of tableaux of bounded height

TL;DR

This paper studies the combinatorial properties of the number of standard Young tableaux with bounded height, introducing matrices that count such tableaux and analyzing their recursive and asymptotic behaviors.

Contribution

It introduces a new family of matrices counting tableaux with bounded columns and derives their recursive and asymptotic properties.

Findings

Matrices satisfy a three-term recurrence

Recursive formulas for total tableaux count

Asymptotic behavior of tableaux numbers

Abstract

We introduce an infinite family of lower triangular matrices $\Gamma^{(s)}$, where $\gamma_{n,i}^s$ counts the standard Young tableaux on $n$ cells and with at most $s$ columns on a suitable subset of shapes. We show that the entries of these matrices satisfy a three-term row recurrence and we deduce recursive and asymptotic properties for the total number $\tau_s(n)$ of tableaux on $n$ cells and with at most $s$ columns.