Asymptotics of Smallest Component Sizes in Decomposable Combinatorial Structures of Alg-Log Type.

A decomposable combinatorial structure consists of simpler objects called components which by themselves can not be further decomposed. We focus on the multi-set construction where the component generating function C(z) is of alg-log type, that is, C(z) behaves like c + d(1 - z/ρ)α (ln 1 / 1 - z/ρ)β (1 + o(1)) when z is near the dominant singularity ρ. We provide asymptotic results about the size of the smallest components in random combinatorial structures for the cases 0 < α < 1 and any β and α < 0 and β = 0. The particular case α = 0 and β = 1, the so-called exp-log class, has been treated in previous papers. We also provide similar asymptotic estimates for combinatorial objects with a restricted pattern, that is, when part of its factorization pattern is known. We extend our results to include certain type of integers partitions. [ABSTRACT FROM AUTHOR]