The area above the Dyck path of a permutation

TL;DR

This paper explores a mapping from permutations to Dyck paths, analyzing the resulting diagrams and their statistics, and establishing connections to permutation properties and poset structures.

Contribution

It introduces a new filling method for diagrams that uniquely determines permutations and relates diagram containment to the strong Bruhat poset.

Findings

Distribution of diagram sizes analyzed

Generalization of Simion and Schmidt's parity result

Diagram filling method uniquely identifies permutations

Abstract

In this paper we study a mapping from permutations to Dyck paths. A Dyck path gives rise to a (Young) diagram and we give relationships between statistics on permutations and statistics on their corresponding diagrams. The distribution of the size of this diagram is discussed and a generalisation given of a parity result due to Simion and Schmidt. We propose a filling of the diagram which determines the permutation uniquely. Diagram containment on a restricted class of permutations is shown to be related to the strong Bruhat poset.