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Super-strong Wilf equivalence classes of the symmetric group ${\mathcal S}_n$on $n$ letters, with respect to the generalized factor order, were shown byHadjiloucas, Michos and Savvidou (2018) to be in bijection with pyramidalsequences of consecutive differences. In this article we enumerate the latterby giving recursive formulae in terms of a two-dimensional analogue ofnon-interval permutations. As a by-product, we obtain a recursively defined setof representatives of super-strong Wilf equivalence classes in ${\mathcalS}_n$. We also provide a connection between super-strong Wilf equivalence andthe geometric notion of shift equivalence---originally defined by Fidler,Glasscock, Miceli, Pantone, and Xu (2018) for words---by showing that analternate way to characterize super-strong Wilf equivalence for permutations isby keeping only rigid shifts in the definition of shift equivalence. Thisallows 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).

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