Combinatorics of diagrams of permutations

TL;DR

This paper explores combinatorial objects related to permutation diagrams, establishing their equinumerosity and providing a $q$-analogue for counting invertible matrices avoiding certain permutation patterns.

Contribution

It introduces new connections between various combinatorial objects associated with permutations and extends existing results to non-Grassmannian permutations using $q$-analogues.

Findings

Acyclic orientations, rook placements, and fillings of permutation diagrams are equinumerous.

A $q$-analogue relates the Poincaré polynomial to counting invertible matrices over finite fields.

The work generalizes known combinatorial results to broader classes of permutations.

Abstract

There are numerous combinatorial objects associated to a Grassmannian permutation $w_\lambda$ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their $q$-analogues in the case of permutations $w$ that are not necessarily Grassmannian. We give two main results: first, we show that certain acyclic orientations, rook placements avoiding a diagram of $w$, and fillings of a diagram of $w$ are equinumerous for all permutations $w$. Second, we give a $q$-analogue of a result of Hultman-Linusson-Shareshian-Sj\"ostrand by showing that under a certain pattern condition the Poincar\'e polynomial for the Bruhat interval of $w$ essentially counts invertible matrices avoiding a diagram of $w$ over a finite field. In addition to our main results, we include at the end a number of open questions.