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Extreme Values of Permutation Statistics

Philip Dörr, Thomas Kahle

TL;DR

The article addresses extreme values of Mahonian and Eulerian statistics on finite Coxeter groups by constructing row-wise triangular arrays of independent samples across increasing ranks. By decomposing inversion and descent counts into independent summands via generating-function factorizations and applying advanced large-deviation and Berry–Esseen techniques, the authors establish Gumbel-type limit laws for the maxima under precise growth conditions on the row size $k_n$. They obtain detailed EVLT results for sequences of classical Weyl groups, arbitrary finite Coxeter groups, products of Weyl groups, and dihedral compounds, and prove a universal extreme-value theorem that applies under a Berry–Esseen framework with sublinear growth of $k_n$. These results illuminate when extremes of permutation statistics converge to the Gumbel distribution and offer a roadmap for extending extreme-value theory to structured combinatorial settings with dependent statistics.

Abstract

We investigate extreme values of Mahonian and Eulerian distributions arising from counting inversions and descents of random elements of finite Coxeter groups. To this end, we construct a triangular array of either distribution from a sequence of Coxeter groups with increasing ranks. To avoid degeneracy of extreme values, the number of i.i.d. samples $k_n$ in each row must be asymptotically bounded. We employ large deviations theory to prove the Gumbel attraction of Mahonian and Eulerian distributions. It is shown that for the two classes, different bounds on $k_n$ ensure this.

Extreme Values of Permutation Statistics

TL;DR

The article addresses extreme values of Mahonian and Eulerian statistics on finite Coxeter groups by constructing row-wise triangular arrays of independent samples across increasing ranks. By decomposing inversion and descent counts into independent summands via generating-function factorizations and applying advanced large-deviation and Berry–Esseen techniques, the authors establish Gumbel-type limit laws for the maxima under precise growth conditions on the row size . They obtain detailed EVLT results for sequences of classical Weyl groups, arbitrary finite Coxeter groups, products of Weyl groups, and dihedral compounds, and prove a universal extreme-value theorem that applies under a Berry–Esseen framework with sublinear growth of . These results illuminate when extremes of permutation statistics converge to the Gumbel distribution and offer a roadmap for extending extreme-value theory to structured combinatorial settings with dependent statistics.

Abstract

We investigate extreme values of Mahonian and Eulerian distributions arising from counting inversions and descents of random elements of finite Coxeter groups. To this end, we construct a triangular array of either distribution from a sequence of Coxeter groups with increasing ranks. To avoid degeneracy of extreme values, the number of i.i.d. samples in each row must be asymptotically bounded. We employ large deviations theory to prove the Gumbel attraction of Mahonian and Eulerian distributions. It is shown that for the two classes, different bounds on ensure this.
Paper Structure (10 sections, 13 theorems, 54 equations)

This paper contains 10 sections, 13 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries about Coxeter groups and permutation statistics
  3. Tail equivalence for non-i.d. sums
  4. Results for extremes of permutation statistics
  5. Sequences of classical Weyl groups
  6. Arbitrary finite Coxeter groups
  7. Sequences of products of classical Weyl groups.
  8. Sequences involving dihedral groups
  9. Universal extreme value limit theorem for triangular arrays
  10. Outlook

Key Result

Theorem 3

Let $W$ be a finite Coxeter group with $\mathrm{rk}(W) = n$. Then, where $d_1, \ldots, d_n$ are the degrees of $W$.

Theorems & Definitions (25)

  • Definition 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Remark 6
  • Theorem 7: cf. Cramér cramer1938nouveau
  • Theorem 8: cf. Feller feller1943generalization
  • Remark 9
  • Theorem 10: see Petrov & Robinson petrov2008large, Theorem 2.1
  • ...and 15 more