Extremes of generalized inversions on permutation groups

The classical inversion statistic on symmetric groups is the sum of all indicators 1\π(i) > π(j)\ for a random permutation π = (π(1), …, π(n)) and the pairs $(i,j)$ with 1 ≤ i < j ≤ n. The descent statistic counts all i ∈ \1, …, n-1\ with π(i) > π(i+1). The number of inversions can be generalized by restricting the indicators to pairs $(i,j)$ with $i<j$ and |i-j| ≤ d for some d ∈ \1, …, n-1\. Likewise, the number of descents can be generalized by counting all i ∈ \1, …, n-d\ with π(i) > π(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.