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September 15, 2020

Statistics for partitions and unimodal sequences

The study of the asymptotic distribution of statistics for partitions lies at a crossroads of classical methods and the more recent probabilistic framework of Fristedt and others. We discuss two results—one that uses the probabilistic machinery and one that calls for a more direct “elementary” method. We first review Fristedt’s conditioning device and, following Romik, implement a similar construction to give an asymptotic formula for distinct parts partitions of n with largest part bounded by tn−−√. We discuss the intuitive advantages of this approach over a classical circle method/saddle-point method proof. We then turn to unimodal sequences, a generalization of partitions where parts are allowed to increase and then decrease. We use an elementary approach to prove limit shapes for the diagrams of strongly, semi-strict and unrestricted unimodal sequences. We also recover a limit shape for overpartitions via a simple transfer.