Work

Joint Extremes of Inversions and Descents of Random Permutations

Year: 2025

Type: article

Abstract: We provide asymptotic theory for the joint distribution of $$X_\textrm{inv}$$ X inv and $$X_\textrm{des}$$ X des , the numbers of inversions and descents of random permutations. Recently, [14] proved that $$X_\textrm{inv}$$ X inv , respectively, $$X_\textrm{des}$$ X des , is in the maximum domain of attraction of the Gumbel distribution. To tackle the dependency between these two permutation statistics, we use Hájek projections and a suitable quantitative Gaussian approximation. We show that $$(X_\textrm{inv}, X_\textrm{des})$$ ( X inv , X des ) is in the maximum domain of attraction of the two-dimensional Gumbel distribution with independent margins. This result can be stated in the broader combinatorial framework of finite Coxeter groups, on which our method also yields the central limit theorem for $$(X_\textrm{inv}, X_\textrm{des})$$ ( X inv , X des ) and various other permutation statistics as a novel contribution. In particular, signed permutation groups with random biased signs and products of classical Weyl groups are investigated.

Source: Journal of Theoretical Probability

DOI: 10.1007/s10959-025-01407-y

Authors: Philip Dörr, Johannes Heiny

Institutions: Ruhr University Bochum, Stockholm University

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,290

Funder: Stockholms Universitet