Dyson’s crank and unimodal compositions

Abstract

The crank is a partition statistic requested by Dyson in 1944 in order to combinatorially prove a Ramanujan congruence for 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.