Random Partitions with Non-negative rth Differences

Let P"r(n) be the set of partitions of n with non-negative rth differences. Let @l be a partition of an integer n chosen uniformly at random among the set P"r(n). Let d(@l) be a positive rth difference chosen uniformly at random in @l. The aim of this work is to show that for every m>=1, the probability that d(@l)>=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.