On the Alternating runs polynomial in type B and D Coxeter Groups

TL;DR

This paper extends Bóna's group action proof to establish divisibility properties of the alternating runs polynomial in type B and D Coxeter groups, generalizing Wilf's original result for the symmetric group.

Contribution

The paper provides a new proof technique for the divisibility of the alternating runs polynomial in types B and D Coxeter groups, expanding Wilf's classical result.

Findings

Proved divisibility of the alternating runs polynomial in type B and D groups.

Extended Bóna's group action proof to Coxeter groups of types B and D.

Generalized Wilf's result beyond the symmetric group.

Abstract

Wilf showed that the the alternating runs polynomial $R_n(t)$ counting the number of permutations in the Symmetric group is divisible by $(1+t)^m$ where $m = \lfloor (n-2)/2 \rfloor$. Recently, B\'{o}na gave a group action based proof. Type B and D analogues of Wilf's result are known. In this note, we extend B\'{o}na's proof to prove the type B and D analogue.