Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns (original) (raw)

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Keywords: Pattern-replacement, Permutations, Equivalence classes

Abstract

We study a family of equivalence relations on SnS_nSn, the group of permutations on nnn 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 ScS_cSc.

When the partition is of S_3S_3S3 and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in all cases left unresolved by past authros. When the partition is of S3S_3S_3 and has two nontrivial parts, each of size two (as do the Knuth and forgotten relations), we enumerate the classes for 13 of the 14 unresolved cases. In two of these cases, enumerations arise which are the same as those yielded by the Knuth and forgotten relations. The reasons for this phenomenon are still largely a mystery.

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How to Cite

Kuszmaul, W. (2013). Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns. The Electronic Journal of Combinatorics, 20(4), #P10. https://doi.org/10.37236/3330