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Counting the Nontrivial Equivalence Classes of \\(S_n\\) under \\(\\{1234,3412\\}\\)-Pattern-Replacement
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Counting the Nontrivial Equivalence Classes of \\(S_n\\) under \\(\\{1234,3412\\}\\)-Pattern-Replacement
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Paper
Counting the Nontrivial Equivalence Classes of \\(S_n\\) under \\(\\{1234,3412\\}\\)-Pattern-Replacement
2020
Overview
We study the \\(\\{1234, 3412\\}\\) pattern-replacement equivalence relation on the set \\(S_n\\) of permutations of length \\(n\\), which is conceptually similar to the Knuth relation. In particular, we enumerate and characterize the nontrivial equivalence classes, or equivalence classes with size greater than 1, in \\(S_n\\) for \\(n \\geq 7\\) under the \\(\\{1234, 3412\\}\\)-equivalence. This proves a conjecture by Ma, who found three equivalence relations of interest in studying the number of nontrivial equivalence classes of \\(S_n\\) under pattern-replacement equivalence relations with patterns of length \\(4\\), enumerated the nontrivial classes under two of these relations, and left the aforementioned conjecture regarding enumeration under the third as an open problem.
Publisher
Cornell University Library, arXiv.org
Subject
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