Homomesy of Alignments in Perfect Matchings

We investigate the existence of a group action τ that is homomesic with respect to alignments, a type of statistic in perfect matchings. Homomesy is defined as the consistency of an average, and perfect matchings are defined as the set of all partitions of 1 to 2n into pairs. We take advantage of the bijection between labeled Dyck paths and perfect matchings to investigate to investigate the possibility of defining τ inductively. We also relate perfect matchings to the oscillating tableaux and discover relationships that support the possibility of the existence of τ but do not prove it. We find some surprisingly clean formulae for statistical averages of the analogue to the number of alignments for oscillating tableaux of arbitrary shape, which suggests that τ may exist for more general contexts than just perfect matchings.