On super-strong Wilf equivalence classes of permutations

TL;DR

This paper characterizes super-strong Wilf equivalence classes of permutations, providing necessary and sufficient conditions and proving that each class size is a power of two.

Contribution

It offers a complete characterization of super-strong Wilf equivalence using distances and two-colored binary trees, confirming a conjecture about class sizes.

Findings

Necessary and sufficient condition for super-strong Wilf equivalence

Characterization of equivalence classes via two-colored binary trees

Proof that class sizes are powers of two

Abstract

Super-strong (elsewhere referred to as strong) Wilf equivalence is a type of Wilf equivalence on words that was introduced by Kitaev et al. in 2009. We provide a necessary and sufficient condition for two permutations in $n$ letters to be super-strongly Wilf equivalent, using distances between letters within a permutation. Furthermore, we give a characterization of such equivalence classes via two-colored binary trees. This allows us to prove, in the case of super-strong Wilf equivalence, the conjecture stated in (Kitaev et al., 2009) that the cardinality of each Wilf equivalence class is a power of 2.