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Multinomial random combinatorial structures and $r$-versions of Stirling, Eulerian and Lah numbers

Alexander Iksanov, Zakhar Kabluchko, Alexander Marynych, Vitali Wachtel

Abstract

We introduce multinomial and $r$-variants of several classic objects of combinatorial probability, such as the random recursive and Hoppe trees, random set partitions and compositions, the Chinese restaurant process, Feller's coupling, and some others. Just as various classic combinatorial numbers - like Stirling, Eulerian and Lah numbers - emerge as essential ingredients defining the distributions of the mentioned processes, the so-called $r$-versions of these numbers appear in exact distributional formulas for the multinomial and $r$-counterparts. This approach allows us to offer a concise probabilistic interpretation for various identities involving $r$-versions of these combinatorial numbers, which were either unavailable or meaningful only for specific values of the parameter $r$. We analyze the derived distributions for fixed-size structures and establish distributional limit theorems as the size tends to infinity. Utilizing the aforementioned generalized Stirling numbers of both kinds, we define and analyze $(r,s)$-Lah distributions, which have arisen in the existing literature on combinatorial probability in various contexts.

Multinomial random combinatorial structures and $r$-versions of Stirling, Eulerian and Lah numbers

Abstract

We introduce multinomial and -variants of several classic objects of combinatorial probability, such as the random recursive and Hoppe trees, random set partitions and compositions, the Chinese restaurant process, Feller's coupling, and some others. Just as various classic combinatorial numbers - like Stirling, Eulerian and Lah numbers - emerge as essential ingredients defining the distributions of the mentioned processes, the so-called -versions of these numbers appear in exact distributional formulas for the multinomial and -counterparts. This approach allows us to offer a concise probabilistic interpretation for various identities involving -versions of these combinatorial numbers, which were either unavailable or meaningful only for specific values of the parameter . We analyze the derived distributions for fixed-size structures and establish distributional limit theorems as the size tends to infinity. Utilizing the aforementioned generalized Stirling numbers of both kinds, we define and analyze -Lah distributions, which have arisen in the existing literature on combinatorial probability in various contexts.
Paper Structure (29 sections, 54 theorems, 260 equations)

This paper contains 29 sections, 54 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. r-Stirling numbers
  3. Three families of probability distributions related to the r-Stirling numbers
  4. Notation
  5. Multinomial structures
  6. Multinomial Stirling numbers
  7. Multinomial Ewens permutation
  8. Multinomial Chinese restaurant and the colored Feller coupling
  9. Multinomial Hoppe forests
  10. Expected profiles of multinomial Hoppe forests
  11. Leaves in multinomial Hoppe forests and generalized Eulerian numbers
  12. r-Ewens permutations, random r-recursive trees
  13. r-Ewens permutations
  14. Random r-recursive trees and r-Hoppe trees
  15. Random r-partitions
  16. ...and 14 more sections

Key Result

Proposition 2.1

With $x_1,\dots, x_d$ denoting variables,

Theorems & Definitions (136)

  • Definition 1.1
  • Definition 1.2: nishimura_sibuyasibuya_stirling_familysibuya_nishimura_breaking
  • Definition 1.3
  • Definition 1.4: nishimura_sibuyasibuya_stirling_family
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 126 more