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 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

[1]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[2]  E. Wright.  On the Coefficients of Power Series Having Exponential Singularities , 1933 .

[3]  Leonard Carlitz.  The coefficients of certain power series , 1955 .

[4]  Zhicheng Gao,et al.  Enumeration of Decomposable Combinatorial Structures with Restricted Patterns , 2009 .

[5]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[6]  Cristian S. Calude,et al.  Discrete Mathematics and Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

[7]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[8]  Philippe Flajolet,et al.  Gaussian limit distributions and exponential tails , 1993 .

[9]  L. B. Richmond.  The moments of partitions, II , 1975 .

[10]  Zhicheng Gao,et al.  The Size of the rth Smallest Component in Decomposable Structures with a Restricted Pattern , 2006 .

[11]  Asymptotic relations for partitions , 1976 .

[12]  W. Hayman.  A Generalisation of Stirling's Formula. , 1956 .

[13]  Philippe Flajolet,et al.  Gaussian limiting distributions for the number of components in combinatorial structures , 1990, J. Comb. Theory, Ser. A.

[14]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[15]  Xavier Gourdon,et al.  Largest component in random combinatorial structures , 1998, Discret. Math..

[16]  Andrew D. Barbour,et al.  Random Combinatorial Structures and Prime Factorizations , 1997 .

[17]  Daniel Panario,et al.  Smallest components in decomposable structures: Exp-log class , 2001, Algorithmica.