Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic probability of irreducibles II: sequence

Thierry Monteil, Khaydar Nurligareev

TL;DR

The paper derives a complete asymptotic expansion for the probability that a large SEQ-decomposed object is irreducible or has a given number of SEQ-irreducible parts, for gargantuan labeled and unlabeled classes. It develops a unified framework based on gargantuan sequences, Bender's theorem, and the lift operation to handle non-stable decompositions, and applies it to a broad spectrum of combinatorial constructions including tournaments, multitournaments, permutations, linear orders, and perfect matchings. Key contributions include explicit coefficient formulas d_{k,m} and their unlabeled analogues, and a demonstration of how these coefficients decompose into combinatorial structures via inclusion-exclusion and lifting. The results illuminate when SEQ-based irreducibility probabilities admit combinatorial interpretations and suggest directions for extending the approach to other decompositions and to anti-SEQ formulations.

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.

Asymptotic probability of irreducibles II: sequence

TL;DR

The paper derives a complete asymptotic expansion for the probability that a large SEQ-decomposed object is irreducible or has a given number of SEQ-irreducible parts, for gargantuan labeled and unlabeled classes. It develops a unified framework based on gargantuan sequences, Bender's theorem, and the lift operation to handle non-stable decompositions, and applies it to a broad spectrum of combinatorial constructions including tournaments, multitournaments, permutations, linear orders, and perfect matchings. Key contributions include explicit coefficient formulas d_{k,m} and their unlabeled analogues, and a demonstration of how these coefficients decompose into combinatorial structures via inclusion-exclusion and lifting. The results illuminate when SEQ-based irreducibility probabilities admit combinatorial interpretations and suggest directions for extending the approach to other decompositions and to anti-SEQ formulations.

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.

Paper Structure

This paper contains 25 sections, 29 theorems, 146 equations, 8 figures, 21 tables.

Table of Contents

  1. Introduction
  2. Tools
  3. Asymptotic notation
  4. Combinatorial classes and decompositions
  5. Gargantuan classes and Bender's theorem
  6. Asymptotics of the labeled construction SEQ
  7. Asymptotic probability of labeled SEQ-irreducibles
  8. Combinatorial interpretation of the asymptotic theorem
  9. Asymptotics of irreducible labeled tournaments
  10. Irreducible multitournaments and their asymptotics
  11. Irreducible linear orders and their asymptotics
  12. Lift
  13. Indecomposable permutations
  14. Lift operation
  15. Asymptotics of indecomposable permutations
  16. ...and 10 more sections

Key Result

Theorem \ref{theorem: SEQ_m-asymptotics}

Let $\mathcal{A}$ be a gargantuan labeled combinatorial class satisfying $\mathcal{A} = \mathrm{SEQ}(\mathcal{B})$ for some labeled combinatorial class $\mathcal{B}$. Suppose that $a\in\mathcal{A}$ is a random object of size $n$. In this case, for any positive integer $m$, where Here, $(\mathfrak b_n^{(k)})$ is the counting sequence of the combinatorial class $\mathrm{SEQ}_k(\mathcal{B}) = \math

Figures (8)

  • Figure 1: Schema for counting objects in $\mathcal{A}_n$.
  • Figure 2: Sequences with two small and one large components.
  • Figure 3: Specific sequences with three small and one large components.
  • Figure 4: Specific sequence with four small and one large components.
  • Figure 5: Relabeling of $\ell = (2 < 1 < 3)$ corresponding to $\rho = \left(123312\right).$
  • ...and 3 more figures

Theorems & Definitions (62)

  • Theorem \ref{theorem: SEQ_m-asymptotics}: $\mathrm{SEQ}$-asymptotics
  • Definition \ref{theorem: SEQ_m-asymptotics}
  • Definition \ref{theorem: SEQ_m-asymptotics}
  • Definition \ref{theorem: SEQ_m-asymptotics}
  • Theorem \ref{theorem: SEQ_m-asymptotics}: Bender Bender1975
  • Lemma \ref{theorem: SEQ_m-asymptotics}: Lemma 2.4 in MonteilNurligareevSET
  • Lemma \ref{theorem: SEQ_m-asymptotics}: Lemma 2.5 in MonteilNurligareevSET
  • Lemma \ref{theorem: SEQ_m-asymptotics}
  • proof
  • Theorem \ref{theorem: SEQ_m-asymptotics}
  • ...and 52 more