Equivalence classes of permutations avoiding a pattern

TL;DR

This paper explores various equivalence relations on permutations avoiding a pattern, revealing connections to known classes, counting sequences, and introducing new symmetries and Wilf-equivalences.

Contribution

It systematically studies four permutation equivalence relations and uncovers their combinatorial structures, linking them to known sequences and introducing new symmetries.

Findings

Involutions correspond to conjugacy classes.

Knuth-equivalence relates to hook-shaped insertion tableaux.

Toric equivalence classes are counted by Euler's totient and divisor functions.

Abstract

Given a permutation pattern p and an equivalence relation on permutations, we study the corresponding equivalence classes all of whose members avoid p. Four relations are studied: Conjugacy, order isomorphism, Knuth-equivalence and toric equivalence. Each of these produces a known class of permutations or a known counting sequence. For example, involutions correspond to conjugacy, and permutations whose insertion tableau is hook-shaped with 2 in the first row correspond to Knuth-equivalence. These permutations are equinumerous with certain congruence classes of graph endomorphisms. In the case of toric equivalence we find a class of permutations that are counted by the Euler totient function, with a subclass counted by the number-of-divisors function. We also provide a new symmetry for bivincular patterns that produces some new non-trivial Wilf-equivalences