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
Highlights
A decomposable combinatorial structure consists of simpler objects called components which by themselves can not be further decomposed
We assume that the component generating function C(z) is of alg-log type, that is, near the singularity ρ, C(z) behaves like c + d(1 − z/ρ)α ln 1 β
Definition 2.1 The restricted pattern of an object of size n is a mapping S : J → N, where J is a set of components’ sizes, N is the set of nonnegative integers, and S(j) is the number of Asymptotics of Smallest Component: Alg-Log Type components of size j
Summary
A decomposable combinatorial structure consists of simpler objects called components which by themselves can not be further decomposed. The class of generic alg-log component generating functions (that is, not for the particular exp-log case) was introduced by Flajolet and Soria [7] when studying the number of components in combinatorial structures.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
