Random Partitions with Parts in the Range of a Polynomial

Let Ω(n,Q) be the set of partitions of n into summands that are elements of the set A = { Q(k) : k ∈ Z } . Here Q ∈ Z[x] is a fixed polynomial of degree d > 1 which is increasing on R, and such that Q(m) is a non– negative integer for every integer m ≥ 0. For every λ ∈ Ω(n,Q), letMn(λ) be the number of parts, with multiplicity, that λ has. Put a uniform probability distribution on Ω(n,Q), and regard Mn as a random variable. The limiting density of the random variable Mn (suitably normalized) is determined explicitly. 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.