New results on families of pattern-replacement equivalences

Abstract

We study pattern-replacement equivalences of a similar nature to those studied by Linton, Propp, Roby, and West. Given any partition P of Sc, we consider two equivalence relations acting on Sn associated with P, loose P-equivalence and tight P-equivalence.We develop a general framework for studying tight pattern-replacement equivalences. Using our machinery, we count equivalence classes in Sn for three tight equivalences posed as open problems by Linton, Propp, Roby, and West, and we find systems of equivalence-class representatives for two others, also previously posed as open problems. In addition, we study several infinite families of tight equivalences. In particular, we characterize the equivalence classes in Sn under the most general tight equivalence, tight Sc-equivalence.Moreover, we extend past work on loose P-equivalences (for various P) by characterizing equivalence classes under three infinite families of loose P-equivalences. Particularly interesting members of these families include loose {123,132}-equivalence (previously posed as an open problem by Linton, Propp, Roby, and West), and loose {123,132}{321,231}-equivalence. For both equivalences, we enumerate the classes in Sn.