Asymptotic probability of irreducibles II: sequence

TL;DR

This paper investigates the detailed asymptotic behavior of the probability that large combinatorial objects are irreducible or have a fixed number of irreducible components, revealing integer coefficients linked to related combinatorial classes.

Contribution

It provides a comprehensive asymptotic expansion for irreducibility probabilities in large combinatorial structures, with explicit integer coefficients and applications to various classes.

Findings

Coefficients in the expansion are integers.

Coefficients can be expressed as linear combinations of related class counts.

Results apply to labeled and unlabeled combinatorial objects like permutations and matchings.

Abstract

This paper is devoted to the structure of the complete asymptotic expansion of the probability that a large combinatorial object is irreducible or consists of a given number of irreducible parts, where irreducibility is understood in terms of combinatorial construction SEQ, labeled or unlabeled. We show that for rapidly growing (i.e. gargantuan) combinatorial classes, the coefficients that appear in this expansion are integers and can be interpreted as linear combinations of the counting sequences of three closely related combinatorial classes. We apply this general asymptotic result to labeled and unlabeled (multi-)tournaments, as well as to (multi-)permutations and (multi-)matchings. We also explore the limits of our approach with respect to other combinatorial constructions.