Dyson's crank and unimodal compositions

TL;DR

This paper explores the relationship between Dyson's crank statistic and unimodal compositions, providing new combinatorial insights and a proof of a recent Euler pentagonal number theorem.

Contribution

It establishes connections between Dyson's crank and unimodal compositions and offers a combinatorial proof of a new truncated Euler pentagonal number theorem.

Findings

Connections between Dyson's crank and unimodal compositions

A combinatorial proof of Xia and Zhao's Euler pentagonal number theorem

Enhanced understanding of partition statistics

Abstract

The crank is a partition statistic requested by Dyson in 1944 in order to combinatorially prove a Ramanujan congruence of Euler's partition function $p(n)$. In this paper, we provide connections between Dyson's crank and unimodal compositions. Somewhat unrelated, we give a combinatorial proof of a new truncated Euler pentagonal number theorem due to Xia and Zhao.