Homomesies on permutations -- an analysis of maps and statistics in the FindStat database

TL;DR

This paper systematically studies permutation statistics and bijections, identifying 122 instances of homomesy where the average statistic value is consistent across orbits, using computational searches in the FindStat database.

Contribution

It provides the first comprehensive catalog of homomesy phenomena on permutations, combining theoretical proofs with computational methods.

Findings

122 homomesy instances identified and proved

New homomesy results for various permutation statistics

Use of SageMath and FindStat database for discovery

Abstract

In this paper, we perform a systematic study of permutation statistics and bijective maps on permutations in which we identify and prove 122 instances of the homomesy phenomenon. Homomesy occurs when the average value of a statistic is the same on each orbit of a given map. The maps we investigate include the Lehmer code rotation, the reverse, the complement, the Foata bijection, and the Kreweras complement. The statistics studied relate to familiar notions such as inversions, descents, and permutation patterns, and also more obscure constructs. Beside the many new homomesy results, we discuss our research method, in which we used SageMath to search the FindStat combinatorial statistics database to identify potential homomesies.