Distribution of Alternating Sums of Parts in Partitions

TL;DR

This paper analyzes the distribution of the alternating sum statistic in partitions, computing all moments using the Circle Method, and proposes a general framework for studying similar partition statistics.

Contribution

It introduces a comprehensive method for calculating moments of the alternating sum statistic and a new framework for analyzing partition statistics more broadly.

Findings

All moments of the alternating sum statistic are computed.

The Circle Method effectively proves the distribution results.

A flexible framework for future partition statistic studies is proposed.

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.