Congruences for hook lengths of partitions

TL;DR

This paper demonstrates how to derive and generalize congruences for hook lengths in various classes of partitions using addition theorems, extending previous results to new partition types.

Contribution

It introduces a unified approach to derive congruences for hook lengths across multiple partition classes using addition theorems.

Findings

Derived congruences for hook lengths of self-conjugate partitions.

Generalized congruences to all partitions using addition theorems.

Extended congruences to z-asymmetric partitions, including special cases.

Abstract

Recently, Amdeberhan et al. proved congruences for the number of hooks of fixed even length among the set of self-conjugate partitions of an integer $n$, therefore answering positively a conjecture raised by Ballantine et al.. In this paper, we show how these congruences can be immediately derived and generalized from an addition theorem for self-conjugate partitions proved by the second author. We also recall how the addition theorem proved before by Han and Ji can be used to derive similar congruences for the whole set of partitions, which are originally due to Bessenrodt, and Bacher and Manivel. Finally, we extend such congruences to the set of $z$-asymmetric partitions defined by Ayyer and Kumari, by proving an addition-multiplication theorem for these partitions. Among other things, this contains as special cases the congruences for the number of hook lengths for the self-conjugate and the so-called doubled distinct partitions.