Generalizing Eulerian Numbers via Semipermutations: Topological and Combinatorial Aspects

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 first obtained by an algebraic-topological argument, and all of them are explicitly described in combinatorial terms.