Extremes of generalized inversions on symmetric groups

Philip Dörr

Published 2024-04-09Version 1

The classical inversion statistic on symmetric groups is the sum of all indicators $\textbf{1}\{\pi(i) > \pi(j)\}$ for a random permutation $\pi = (\pi(1), \ldots, \pi(n))$ and the pairs $(i,j)$ with $1 \leq i < j \leq n$. The descent statistic counts all $i \in \{1, \ldots, n-1\}$ with $\pi(i) > \pi(i+1)$. The number of inversions can be generalized by restricting the indicators to pairs $(i,j)$ with $i<j$ and $|i-j| \leq d$ for some $d \in \{1, \ldots, n-1\}$. Likewise, the number of descents can be generalized by counting all $i \in \{1, \ldots, n-d\}$ with $\pi(i) > \pi(i+d)$. These generalized statistics can be further extended to the signed and even-signed permutation groups. The bandwidth index $d$ can be chosen in dependence of $n$, and the magnitude of $d$ is significant for asymptotic considerations. It is known that each of these statistics is asymptotically normal for suitable choices of $d$. In this paper we prove the bivariate asymptotic normality and determine the extreme value asymptotics of generalized inversions and descents.