Equivalence Classes in Sn for Three Families of PatternReplacement Relations

ABSTRACT. We study a family of equivalence relations on Sn, the group of permutations on n letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of Sc. In particular, we are interested in the number of classes created in Sn by each relation and in characterizing these classes. Imposing the condition that the partition of Sc has one nontrivial part containing the cyclic shifts of a single permutation, we find enumerations for the number of nontrivial classes. When the per-mutation is the identity, we are able to compare the sizes of these classes and connect parts of the problem to Young tableaux and Catalan lattice paths. Imposing the condition that the partition has one nontrivial part containing all of the permutations in Sc beginning with 1, we both enumerate and characterize the classes in Sn. We do the same for the partition that has two nontrivial parts, one containing all of the permutations in Sc beginning with 1, and one containing all of the permutations in Sc ending with 1. 1