Restricted Grassmannian permutations

TL;DR

This paper studies Grassmannian permutations, focusing on pattern avoidance and parity restrictions, providing enumeration formulas, combinatorial interpretations, and bijections with Dyck and Schr"oder paths.

Contribution

It introduces new enumeration formulas for pattern-avoiding Grassmannian permutations and establishes combinatorial links with Dyck and Schr"oder paths.

Findings

Derived formulas for pattern-avoiding Grassmannian permutations.

Established bijections with Dyck and Schr"oder paths.

Enumerated odd and even Grassmannian permutations.

Abstract

A permutation is called Grassmannian if it has at most one descent. In this paper, we investigate pattern avoidance and parity restrictions for such permutations. As our main result, we derive formulas for the enumeration of Grassmannian permutations that avoid a classical pattern of arbitrary size. In addition, for patterns of the form $k12\cdots(k-1)$ and $23\cdots k1$, we provide combinatorial interpretations in terms of Dyck paths, and for $35124$-avoiding Grassmannian permutations, we give an explicit bijection to certain pattern-avoiding Schr\"oder paths. Finally, we enumerate the subsets of odd and even permutations and discuss properties of their corresponding Dyck paths.