Identically Distributed Pairs of Partition Statistics

TL;DR

The paper demonstrates that many partition theorems asserting equinumerosity are instances of a broader principle where partition statistics have identical distribution functions, extending sieve-equivalence methods.

Contribution

It introduces a general framework showing that equinumerous partition classes correspond to partition statistics with identical distributions, strengthening classical results.

Findings

Many partition theorems are special cases of a stronger distributional equality.

The method extends sieve-equivalence to establish distributional identities.

Provides simple criteria for when partition statistics have identical distributions.

Abstract

We show that many theorems which assert that two kinds of partitions of the same integer $n$ are equinumerous are actually special cases of a much stronger form of equality. We show that in fact there correspond partition statistics $X$ and $Y$ that have identical distribution functions. The method is an extension of the principle of sieve-equivalence, and it yields simple criteria under which we can infer this identity of distribution functions.