Crossings and alignments of permutations

TL;DR

This paper introduces new $q$-analogs of Eulerian numbers related to permutation crossings and alignments, deriving their generating functions, exploring their combinatorial properties, and connecting them to the ASEP model's stationary distribution.

Contribution

It derives the continued fraction form of the generating functions for new $q$-Eulerian numbers and links these to permutation patterns and the ASEP model.

Findings

Derived continued fraction form of generating functions.

Connected $q$-Eulerian numbers to permutation crossings and patterns.

Linked these numbers to the stationary distribution of ASEP.

Abstract

We derive the continued fraction form of the generating function of some new $q$-analogs of the Eulerian numbers $\hat{E}_{k,n}(q)$ introduced by Lauren Williams building on work of Alexander Postnikov. They are related to the number of alignments and weak exceedances of permutations. We show how these numbers are related to crossing and generalized patterns of permutations We generalize to the case of decorated permutations. Finally we show how these numbers appear naturally in the stationary distribution of the ASEP model.