Random Partitions with Non Negative rth Differences

Abstract

Let Pr(n) be the set of partitions of n with non negative r ^th differences. Let λ be a partition of an integer n chosen uniformly at random among the set Pr(n) Let d(λ) be a positive r ^th difference chosen uniformly at random in λ. The aim of this work is to show that for every m ≥ 1, the probability that d(λ) ≥ m approaches the constant m ^?1/r as n → ∞ This work is a generalization of a result on integer partitions [7] and was motivated by a recent identity by Andrews, Paule and Riese’s Omega package [3]. To prove this result we use bijective, asymptotic/analytic and probabilistic combinatorics.