Distribution of Alternating Sums of Parts in Partitions
William Craig, Runqiao Li
TL;DR
The paper studies the distribution of the alternating-sum statistic $a(\lambda)$ across partitions by computing all moments $A_m(n)$ and normalized moments $\mathbb{E}_m(n)$ via generating functions built with MacMahon's partition analysis. It derives a two-variable generating function $P(z;q)=\dfrac{1}{(zq;q^2)_\infty (q^2;q^2)_\infty}$ and applies Euler–Maclaurin summation together with Wright's Circle Method to obtain precise asymptotics for $\mathbb{E}_m(n)$; the main result is $\mathbb{E}_m(n) \sim \dfrac{6^{m/2}}{2^m \pi^m} n^{m/2} \log^m\left( \dfrac{\sqrt{6n}}{\pi} \right)$ as $n\to\infty$. The work also connects to the distribution of the number of odd parts and proposes a general framework for studying distributions of partition statistics using MacMahon's analysis, potentially enabling joint distributions and broader generalizations.
Abstract
Recently, many authors have investigated how various partition statistics distribute as the size of the partition grows. In this work, we look at a particular statistic arising from the recent rejuvenation of MacMahon's partition analysis. More specifically, we compute all the moments of the alternating sum statistic for partitions. We prove this results using the Circle Method. We also propose a general framework for studying further questions of this type that may avoid some of the complications that arise in traditional approaches to the distributions of partition statistics, and we comment on the utility, comparative ease and opportunities to generalize to very broad settings.