Identities for the number of standard Young tableaux in some $(k,\ell)$ hooks

TL;DR

This paper investigates formulas for counting standard Young tableaux within specific hook shapes, extending known results for certain small parameters to broader cases.

Contribution

It derives new formulas for the number of tableaux in $(k, ext{ell})$ hooks, expanding the understanding beyond previously known cases with small parameters.

Findings

Formulas for $S(k, ext{ell};n)$ when $k+ ext{ell} extless=4$

Extension of known counts for tableaux in hook shapes

Enhanced understanding of tableau enumeration in hook constraints

Abstract

Closed formulas are known for $S(k,0;n)$, the number of standard Young tableaux of size $n$ and with at most $k$ parts, where $1\le k\le 5$. Here we study the analogue problem for $S(k,\ell;n)$, the number of standard Young tableaux of size $n$ which are contained in the $(k,\ell)$ hook. We deduce some formulas for the cases $k+\ell\le 4$.