Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities

TL;DR

This paper investigates the distribution of $t$-hooks in partition classes related to classical identities, deriving generating functions and inequalities for specific cases, advancing understanding of partition hook statistics.

Contribution

It introduces new generating functions and inequalities for the number of $t$-hooks in partition classes from Rogers-Ramanujan and Göllitz identities, focusing on cases $t=1,2$.

Findings

Derived generating functions for $t$-hooks.

Proved inequalities for $t=1,2$.

Provided asymptotic formulas for $t$-hook counts.

Abstract

Motivated by recent study on the number of $t$-hooks in partitions arising from Euler's partition identity, we investigate the number of $t$-hooks in the sets from the first Rogers-Ramanujan identity and the first little G\"ollitz identity. In particular, for $t=1,2$, we obtain the generating functions for the number of $t$-hooks and prove $t$-hook inequalities by deriving asymptotic formulas.