A generalization of conjugation of integer partitions

TL;DR

The paper constructs a parameterized family of involutions on partitions of $n$ for each positive integer $s$, swapping the two central statistics $r_s$ (the number of parts divisible by $s$) and $c_s$ (the number of $s$-cells in the Ferrers diagram), thereby proving joint symmetry of their distributions. For $s=1$ the involution reduces to conjugation, while for general $s$ the authors develop a refined framework based on remainder sequences ${\boldsymbol\rho}_s(\lambda)$ and remainder diagrams that fixes this sequence while permuting the pair $(r_s,c_s)$. The involution is built step by step: first in the empty remainder sequence case, then via a crucial removal operation for nonzero remainders, then for strictly increasing remainder sequences, and finally reduced to the general case using a reduction algorithm that passes to a corresponding remainder diagram without yellow cells. Alongside, explicit generating-function expressions are derived: for fixed remainder sequence and $(r,c)$, the generating function in $q$ (with $Q=q^s$) factors into a form that makes the $r$–$c$ symmetry evident, and a corollary rewrites this in a symmetry-manifest form involving $q$-shifted factorials. The paper also provides two proofs of the coefficient formula, one combinatorial and one computational, and connects these constructions to established bijections such as Glaisher–Franklin and Loehr–Warrington, enriching the theory of partition bijections and $q$-series identities.

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.

Figures (15)

• Figure 1: The Ferrers diagrams of $\lambda=(6,4,4,1)$ and $\lambda'=(4,3,3,3,1,1)$
• Figure 2: The partitions $\lambda$ of 37 with remainder sequence $(2,1,1,2,1)$, $r_3(\lambda)=2$ and $c_3(\lambda)=3$
• Figure 3: The partitions $\lambda$ of 37 with remainder sequence $(2,1,1,2,1)$, $r_3(\lambda)=3$ and $c_3(\lambda)=2$
• Figure 4: Example partitions for \ref{['delete']}
• Figure 5: A partition of 74 and its 4-reduction

Theorems & Definitions (30)

• Theorem 1
• Example 2
• Theorem 3
• Corollary 4
• Example 5
• Remark 6
• Lemma 7
• proof
• Lemma 8
• proof