Joint Extremes of Inversions and Descents of Random Permutations

Philip Dörr () and Johannes Heiny ()
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Philip Dörr: Ruhr University Bochum
Johannes Heiny: Stockholm University

Journal of Theoretical Probability, 2025, vol. 38, issue 2, 1-37

Abstract: 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.

Keywords: Permutation statistics; Joint distribution; Extreme value theory; Central limit theorem; Maximum; Coxeter group; Primary 60G70; 62R01; Secondary 20F55; 05A16 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01407-y

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