Geodesic
Asymptotics of Smallest Component Sizes in Decomposable Combinatorial Structures of Alg-Log Type
Discrete mathematics & theoretical computer science, Tome 12 (2010) no. 2.

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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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     author = {Dong, Li and Gao, Zhicheng and Panario, Daniel and Richmond, Bruce},
     title = {Asymptotics of {Smallest} {Component} {Sizes} in {Decomposable} {Combinatorial} {Structures} of {Alg-Log} {Type}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {2010},
     doi = {10.46298/dmtcs.503},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.503/}
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Dong, Li; Gao, Zhicheng; Panario, Daniel; Richmond, Bruce. Asymptotics of Smallest Component Sizes in Decomposable Combinatorial Structures of Alg-Log Type. Discrete mathematics & theoretical computer science, Tome 12 (2010) no. 2. doi : 10.46298/dmtcs.503. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.503/

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