Statistics for Unimodal Sequences

Walter Bridges, Kathrin Bringmann

Published 2021-06-04Version 1

We prove a number of limiting distributions for statistics for unimodal sequences of positive integers by adapting a probabilistic framework for integer partitions introduced by Fristedt. The difficulty in applying the direct analogue of Fristedt's techniques to unimodal sequences lies in the fact that the generating function for partitions is an infinite product, while that of unimodal sequences is not. Essentially, we get around this by conditioning on the size of the largest part and working uniformly on contributing summands. Our framework may be used to derive many distributions, and our results include joint distributions for largest parts and multiplicities of small parts. We discuss ranks as well. We further obtain analogous results for strongly unimodal sequences.