Random partitions with non negative rth differences

Abstract

Let $P_r(n)$ be the set of partitions of n with non negative rth differences. Let $λ$ be a partition chosen uniformly at random among the set $P_r(n)$. Let $d(λ)$ be a positive rth difference chosen uniformly at random in $λ$. The aim of this work is to show that for every $m\ge 1$, the probability that $d(λ)\ge m$ approaches $m^{-1/r}$ as $n\to\infty$. To prove this result we use bijective, asymptotic/analytic, and probabilistic combinatorics.