Krystian Gajdzica, Jagiellonian University

The partition function p(n) counts the number of representations of n as sums of positive integers, where the order of summands does not matter. The history of the partition function goes back to Euler, who among other things discovered its generating function. In the recent literature concerning the theory of partitions, there is wealth of papers devoted to log-behaviour of various partition statistics. In particular, the so-called log-concavity property enjoys great popularity these days.

A sequence (a_n)_n≥0 of positive real numbers is said to be log-concave if the inequality a_n² > a_n−1a_n+1 holds for all but finitely many numbers n. In this talk, we discuss various perspectives on that issue, and its generalizations in the context of the partition theory. Essentially, we focus on that property for the A-partition function p_A(n), which enumerates representations of n as sums of numbers from a fixed multiset A of positive integers.

The talk is based on my PhD thesis and my recent joint work with Bernhard Heim and Markus Neuhauser.