An equidistribution involving invisible inversions

TL;DR

This paper introduces two explicit bijections showing that the distribution of invisible inversions in permutations matches the distribution of a specific vincular pattern after sorting runs.

Contribution

It provides a novel combinatorial proof linking invisible inversions with vincular pattern occurrences through explicit bijections.

Findings

Invisible inversions are equidistributed with 13-2 pattern occurrences.

Explicit bijections are constructed to demonstrate this equidistribution.

The results deepen understanding of permutation pattern distributions.

Abstract

We provide two explicit bijections demonstrating that, among permutations, the number of invisible inversions is equidistributed with the number of occurrences of the vincular pattern 13-2 after sorting the set of runs.