Generalizing Eulerian Numbers via Semipermutations: Topological and Combinatorial Aspects

TL;DR

This paper introduces a generalized class of Eulerian numbers through semipermutations, exploring their topological and combinatorial properties, and establishing bijections with symmetric group subsets that preserve key statistics.

Contribution

It extends Eulerian numbers via semipermutations, linking algebraic topology and combinatorics, and identifies symmetric group subsets in bijection with these semipermutations.

Findings

Betti numbers generalize Eulerian numbers

Three symmetric group subsets are in bijection with semipermutations

Bijections preserve lec and des statistics

Abstract

In a paper by Lin an interesting family of semipermutations comes out to index the elements of a cohomology basis of a Hessenberg type variety. The corresponding Betti numbers are a generalization of Eulerian numbers. We show three different subsets of the symmetric group that are in bijection with the set of these semipermutations. These bijections preserve the statistics lec and des: one of these is obtained by an algebraic-topological argument, the others are explicitly described in combinatorial terms.