Rowmotion on 321-avoiding permutations

We give a natural definition of rowmotion for 321-avoiding permutations, by translating, through bijections involving Dyck paths and the Lalanne-Kreweras involution, the analogous notion for antichains of the positive root poset of type A. We prove that some permutation statistics, including the number of fixed points, are homomesic under rowmotion, meaning that they have a constant average over its orbits. We also show that the Armstrong-Stump-Thomas equivariant bijection between order ideals in type A and non-crossing matchings can be described in terms of the Robinson–Schensted–Knuth correspondence on permutations. This is joint work with Ben Adenbaum.