Year: 2010
Type: preprint
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
Source: arXiv (Cornell University)
Author: Ulfarsson, Henning
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1673563934