Extremes of generalized inversions on permutation groups

2024/04/09 by Philip Dörr, Dörr, Philip
Biochemistry, Genetics and Molecular Biology · Computer Science · #05A16 #Algorithms and Data Compression #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Genomic variations and chromosomal abnormalities #Primary: 60G70 #Secondary: 20F55

paper · pdf · doi:10.48550/arxiv.2404.06598

openalex publication_date 2024/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Generalized inversions X_inv^(d) and generalized descents X_des^(d) are an interesting combinatorial extension of the common inversion and descent statistics. By means of the root poset, they can be defined on all classical Weyl groups. In this paper, we investigate the bivariate normality of (X_inv^(d), X_des^(d))^\top as well as the extreme value behavior of X_inv^(d₁), X_des^(d₂) and (X_inv^(d₁), X_des^(d₂))^\top. We show that bivariate normality holds in the regimes of d₁ = o(n^1/3) and d₁ = ω(n^1/2). For these situations, we also discuss the number of samples k_n for which the Gumbel max-attraction applies to a triangular array based on X_inv^(d₁), X_des^(d₂) or (X_inv^(d₁), X_des^(d₂))^\top.

Related

• Joint extremes of inversions and descents of random permutations
• Extreme Values of Permutation Statistics
• Central limit theorems for generalized descents and generalized inversions in finite root systems
• Permutation Statistics in Conjugacy Classes of the Symmetric Group
• Inversions in Colored Permutations, Derangements, and Involutions