Poincar\'e Polynomials of Odd Diagram Classes

TL;DR

This paper proves that the Poincaré polynomial of any odd diagram class factors into simple polynomials, confirming a conjecture that such classes form rank-symmetric intervals in the Bruhat order.

Contribution

The authors present a uniform partitioning algorithm for odd diagram classes and prove their Poincaré polynomials factor into specific simple polynomials, resolving a conjecture.

Findings

Poincaré polynomial factors into polynomials of the form 1+t+...+t^m

Odd diagram classes are rank-symmetric intervals in the Bruhat order

Algorithm for uniform partitioning of odd diagram classes

Abstract

An odd diagram class is a set of permutations with the same odd diagram. Brenti, Carnevale and Tenner showed that each odd diagram class is an interval in the Bruhat order. They conjectured that such intervals are rank-symmetric. In this paper, we present an algorithm to partition an odd diagram class in a uniform manner. As an application, we obtain that the Poincar\'e polynomial of an odd diagram class factors into polynomials of the form $1+t+\cdots+t^m$. This in particular resolves the conjecture of Brenti, Carnevale and Tenner.