  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.
