Regenerative Composition Structures: Characterisation and Asymptotics of Block Counts

Abstract

A regenerative composition structure is a sequence (ℓ _n) of random compositions of integers n = 1, 2,... which satisfies two conditions:

• sampling consistency: if n identical balls are distributed into an ordered series of boxes according to (ℓ _n), then a distributional copy of ℓ _n-1 is obtained by discarding one of the balls picked uniformly at random, and then deleting an empty box in case one is created.

• the first-part deletion invariance: for all n > r > 1, given that the first part of ℓ_n is r, the remaining composition of n — r is distributed like ℓ _n-r.