Equidistributed statistics on matchings and permutations

TL;DR

This paper establishes a new equidistribution between certain statistics in matchings and permutations, providing a non-commutative generating function and proving two conjectures, with potential for further generalizations.

Contribution

It introduces a novel equidistribution result linking matchings and permutations, along with a non-commutative generating function that connects to Fishburn numbers and proves existing conjectures.

Findings

Equidistribution between right nestings/right crossings and permutation patterns.

Non-commutative generating function related to Fishburn numbers.

Proofs of two conjectures by Claesson and Linusson.

Abstract

We show that the bistatistic of right nestings and right crossings in matchings without left nestings is equidistributed with the number of occurrences of two certain patterns in permutations, and furthermore that this equidistribution holds when refined to positions of these statistics in matchings and permutations. For this distribution we obtain a non-commutative generating function which specializes to Zagier's generating function for the Fishburn numbers after abelianization. As a special case we obtain proofs of two conjectures of Claesson and Linusson. Finally, we conjecture that our results can be generalized to involving left crossings of matchings too.