Random Partitions with Non-negative rth Differences

Abstract

Let Pr(n) be the set of partitions of n with non-negative rth 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 rth 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 and was motivated by a recent identity from the Omega package of G. E. Andrews et al. (European J. Combin., MacMahon's partition analysis. III. The Omega package). To prove this result we use bijective, asymptotic/analytic, and probabilistic combinatorics.