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

Abstract : A decomposable combinatorial structure consists of simpler objects called components which by thems elves cannot 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/rho)(alpha) (ln1/1-z/rho)(beta) (1 + o(1)) when z is near the dominant singularity rho. We provide asymptotic results about the size of thes mallest components in random combinatorial structures for the cases 0 < alpha < 1 and any beta, and alpha < 0 and beta=0. The particular case alpha=0 and beta=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 patterns is known. We extend our results to include certain type of integers partitions. partitions
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Lihua Dong, Zhicheng Gao, Daniel Panario, Bruce Richmond. Asymptotics of Smallest Component Sizes in Decomposable Combinatorial Structures of Alg-Log Type. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2010, 12 (2), pp.197-222. ⟨hal-00990464⟩

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