Dyck paths, binary words, and Grassmannian permutations avoiding an increasing pattern

TL;DR

This paper investigates pattern avoidance in Grassmannian permutations, especially avoiding increasing patterns, by relating them to Dyck paths and binary words, and solves a conjecture on counting such permutations.

Contribution

It counts Grassmannian permutations avoiding an increasing pattern, including special classes, and confirms a conjecture by Weiner using combinatorial bijections.

Findings

Counted Grassmannian permutations avoiding the identity pattern.

Established bijections between permutations, Dyck paths, and binary words.

Solved a conjecture on permutation enumeration.

Abstract

A permutation is called Grassmannian if it has at most one descent. The study of pattern avoidance in such permutations was initiated by Gil and Tomasko in 2021. We continue this work by studying Grassmannian permutations that avoid an increasing pattern. In particular, we count the Grassmannian permutations of size $m$ avoiding the identity permutation of size $k$, thus solving a conjecture made by Weiner. We also refine our counts to special classes such as odd Grassmannian permutations and Grassmannian involutions. We prove most of our results by relating Grassmannian permutations to Dyck paths and binary words.