Enumeration of super-strong Wilf equivalence classes of permutations in the generalized factor order

TL;DR

This paper enumerates super-strong Wilf equivalence classes of permutations using recursive formulas, introduces a new set of representatives, and connects these classes to shift equivalence, providing a complete classification and enumeration.

Contribution

It introduces recursive formulas for enumeration, constructs representatives of equivalence classes, and links super-strong Wilf equivalence to shift equivalence for permutations.

Findings

Recursive enumeration formulas for super-strong Wilf classes

A set of representatives for these classes

Complete description and enumeration of shift equivalence classes

Abstract

Super-strong Wilf equivalence classes of the symmetric group ${\mathcal S}_n$ on $n$ letters, with respect to the generalized factor order, were shown by Hadjiloucas, Michos and Savvidou (2018) to be in bijection with pyramidal sequences of consecutive differences. In this article we enumerate the latter by giving recursive formulae in terms of a two-dimensional analogue of non-interval permutations. As a by-product, we obtain a recursively defined set of representatives of super-strong Wilf equivalence classes in ${\mathcal S}_n$. We also provide a connection between super-strong Wilf equivalence and the geometric notion of shift equivalence---originally defined by Fidler, Glasscock, Miceli, Pantone, and Xu (2018) for words---by showing that an alternate way to characterize super-strong Wilf equivalence for permutations is by keeping only rigid shifts in the definition of shift equivalence. This allows us to fully describe shift equivalence classes for permutations of size $n$ and enumerate them, answering the corresponding problem posed by Fidler, Glasscock, Miceli, Pantone, and Xu (2018).