A generalization of conjugation of integer partitions

TL;DR

The paper introduces a family of involutions on integer partitions that demonstrate symmetry in specific statistics, generalizing conjugation and providing explicit generating functions.

Contribution

It presents a new class of involutions for any positive integer parameter, extending the concept of conjugation and connecting to known bijections like the Glaisher-Franklin bijection.

Findings

Involutions show symmetry of partition statistics for any s

Explicit formulas for bivariate generating functions

Generalizes conjugation and relates to known bijections

Abstract

We exhibit, for any positive integer parameter $s$, an involution on the set of integer partitions of $n$. These involutions show the joint symmetry of the distributions of the following two statistics. The first counts the number of parts of a partition divisible by $s$, whereas the second counts the number of cells in the Ferrers diagram of a partition whose leg length is zero and whose arm length has remainder $s-1$ when dividing by $s$. In particular, for $s=1$ this involution is just conjugation. Additionally, we provide explicit expressions for the bivariate generating functions. Our primary motivation to construct these involutions is that we know only of two other "natural" bijections on integer partitions of a given size, one of which is the Glaisher-Franklin bijection sending the set of parts divisible by $s$, each divided by $s$, to the set of parts occurring at least $s$ times.