Extreme Values of Permutation 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.

15 pages, comments welcome, v2: more detailed analysis of k_n, 18 pages, v3: final version. Numbering of statements adjusted to match published version, includes minor improvements over published version (see footnotes)

References in corpus (6)

• A new method of normal approximation
• Counting inversions and descents of random elements in finite Coxeter groups
• A central limit theorem for descents of a Mallows permutation and its inverse
• A central limit theorem for the two-sided descent statistic on Coxeter groups
• Central limit theorem for descents in conjugacy classes of $S_n$
• Joint extremes of inversions and descents of random permutations