Generalizing Eulerian numbers via semipermutations : topological and combinatorial aspects

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.