Random partitions with restricted part sizes

Abstract

AbstractFor a subset $\cal{S}$ of positive integers let Ω(n,$\cal{S}$) be the set of partitions of n into summands that are elements of $\cal{S}$. For every λ ∈ Ω(n,$\cal{S}$), let Mn(λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on Ω(n,$\cal{S}$), and regard Mn as a random variable. In this paper the limiting density of the (suitably normalized) random variable Mn is determined for sets that are sufficiently regular. In particular, our results cover the case $\cal{S}$ = {Q(k) : k ≥ 1}, where Q(x) is a fixed polynomial of degree d ≥ 2. For specific choices of Q, the limiting density has appeared before in rather different contexts such as Kingman's coalescent, and processes associated with the maxima of Brownian bridge and Brownian meander processes. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008