Refined Asymptotics and Explicit Recurrences for the numbers of Young tableaux in the (k,l) hook for k+l less than six

TL;DR

This paper explores refined asymptotics for the number of Young tableaux in small (k,l) hooks, confirming Berele-Regev's leading asymptotics with explicit recurrences and constants using experimental semi-rigorous methods.

Contribution

It introduces explicit recurrences and refined asymptotics for Young tableaux counts in small hooks, extending prior asymptotic results with computational methods.

Findings

Confirmed Berele-Regev asymptotic formula

Derived explicit recurrences for small (k,l) hooks

Computed constants with high accuracy

Abstract

This is an etude in experimental semi-rigorous (rigorizable!) mathematics. The leading asymptotics was brilliantly derived by Allan Berele and Amitai Regev for general hooks H(k,l) and general powers z, but what about more refined asymptotics? For small k and l, one can "guess" a linear recurrence (since we live in the holonomic ansatz) and using the Birkhoff-Trjitzinsky method, beautifully implemented in Doron Zeilberger's Maple package AsyRec (that has been incorporated into the present Maple package), we computed amazing refined asymptotics, that confirm, with a vengeance, the Berele-Regev asymptotic formula, and especially the impressive constant in front!