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Generalizing Eulerian Numbers via Semipermutations: Topological and Combinatorial Aspects

Giovanni Gaiffi, Giovanni Interdonato

TL;DR

This work links the Betti-number structure of Hessenberg-type varieties to generalized Eulerian numbers by analyzing semipermutations that index cohomology bases. It develops three explicit, statistic-preserving bijections between permutations with $i$ descents and subsets of $\\mathcal{S}_n$ characterized by the $\\mathsf{lec}$ statistic and hook/wave-length constraints, including a topological construction and two combinatorial ones (one Foata–Han flavored). These bijections illuminate the combinatorial anatomy of the generalized Eulerian numbers, relate diverse permutation statistics, and unify several counting perspectives via Des/lec correspondences. The results have implications for understanding Hessenberg varieties, their cohomology bases, and the broader combinatorial landscape around Eulerian-type statistics in permutation groups.

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.

Generalizing Eulerian Numbers via Semipermutations: Topological and Combinatorial Aspects

TL;DR

This work links the Betti-number structure of Hessenberg-type varieties to generalized Eulerian numbers by analyzing semipermutations that index cohomology bases. It develops three explicit, statistic-preserving bijections between permutations with descents and subsets of characterized by the statistic and hook/wave-length constraints, including a topological construction and two combinatorial ones (one Foata–Han flavored). These bijections illuminate the combinatorial anatomy of the generalized Eulerian numbers, relate diverse permutation statistics, and unify several counting perspectives via Des/lec correspondences. The results have implications for understanding Hessenberg varieties, their cohomology bases, and the broader combinatorial landscape around Eulerian-type statistics in permutation groups.

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.
Paper Structure (6 sections, 11 theorems, 60 equations)

This paper contains 6 sections, 11 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Some preliminary notation
  3. A bijection pointed out by algebraic topology
  4. An explicit bijection of Sn preserving des-lec statistic: reverse waves
  5. An explicit bijection in the general case of permutations of k+1 different numbers
  6. Second bijection (Foata--Han) preserving des-lec statistic: waves

Key Result

Theorem 3.1

[see Lin24] The $2i$-th Betti number of $\mathcal{X}_{k}$ is given by

Theorems & Definitions (47)

  • Definition 1.1
  • Example 1.2
  • Example 2.1
  • Remark 2.2
  • Example 2.3
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Remark 4.1
  • ...and 37 more