Eulerian Central Limit Theorems and Carlitz identities in positive elements of Classical Weyl Groups

TL;DR

This paper extends Central Limit Theorems and Carlitz identities for descent and excedance statistics from symmetric and alternating groups to positive elements in classical Weyl groups, using signed enumeration methods.

Contribution

It introduces new CLTs for excedances in the alternating group and extends these results to positive elements in types B and D Coxeter groups, refining previous enumerations.

Findings

Central Limit Theorem for excedances in A_n

Extension of CLTs to type B and D Coxeter groups

Refined enumeration of positive and negative elements in D_n

Abstract

Central Limit Theorems are known for the Eulerian statistic "descent" (or "excedance") in the symmetric group $\SSS_n$. Recently, Fulman, Kim, Lee and Petersen gave a Central Limit Theorem for "descent" over the alternating group $\AAA_n$ and also gave a Carlitz identity in $\AAA_n$ using descents. In this paper, we give a Central Limit Theorem in $\AAA_n$ involving excedances. We extend these to the positive elements in type B and type D Coxeter groups. Boroweic and M\l{}otkowski enumerated type B descents over $\DD_n$, the type D Coxeter group and gave similar results. We refine their results for both the positive and negative part of $\DD_n$. Our results are a consequence of signed enumeration over these subsets.