We performed a systematic study of permutation statistics and
bijective maps on permutations, looking for the homomesy phenomenon.
Homomesy occurs when the average value of a statistic is the same on
each orbit of a given map. The maps that exhibit homomesy include the
Lehmer code rotation, the reverse, the complement, and the
Kreweras complement, all of which have some geometric interpretations.
The statistics studied relate to familiar notions such as inversions,
descents, and permutation patterns, among others. Beside the many new
homomesy results, I’ll discuss our research method, in which we used
SageMath to search the FindStat combinatorial statistics database to
identify potential instances of homomesy, and what this experiment
taught us about the maps themselves and the homomesy phenomenon at large.
This is joint work with Jennifer Elder, Erin McNicholas, Jessica Striker
and Amanda Welch.