Polynomial analogues of Ramanujan congruences for Han's hooklength formula

TL;DR

This paper explores polynomial analogues of Ramanujan congruences related to Han's hooklength formula, demonstrating equidistribution properties of coefficients modulo 5 and other primes, and discussing broader symmetries and open questions.

Contribution

It establishes new polynomial congruence results for coefficients in eta powers, extending Ramanujan-like congruences to a broader algebraic setting.

Findings

Coefficients exhibit equidistribution mod 5 for n=5j+4

Symmetries are proved for other primes and prime powers

Open questions on further symmetry properties are raised

Abstract

This article considers the eta power $\prod {(1-q^k)}^{b-1}$. It is proved that the coefficients of $\frac{q^n}{n!}$ in this expression, as polynomials in $b$, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when $n=5j+4$. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.