Permutation Tableaux and the Dashed Permutation Pattern 32-1

TL;DR

This paper establishes a connection between permutation tableaux and the dashed pattern 32-1 in permutations, introducing the inversion number and linking it to pattern occurrences.

Contribution

It introduces the inversion number of permutation tableaux and relates it to the occurrence of the dashed pattern 32-1, solving a problem posed by Corteel and Nadeau.

Findings

Inversion number equals the pattern occurrence in the reverse complement of the permutation.

Permutation tableaux without inversions are characterized as L-Bell tableaux.

The paper provides a bijective proof linking tableaux and permutation patterns.

Abstract

We give a solution to a problem posed by Corteel and Nadeau concerning permutation tableaux of length n and the number of occurrences of the dashed pattern 32--1 in permutations on [n]. We introduce the inversion number of a permutation tableau. For a permutation tableau T and the permutation $\pi$ obtained from T by the bijection of Corteel and Nadeau, we show that the inversion number of T equals the number of occurrences of the dashed pattern 32--1 in the reverse complement of $\pi$. We also show that permutation tableaux without inversions coincide with L-Bell tableaux introduced by Corteel and Nadeau.