Abstract
Super-strong Wilf equivalence is a type of Wilf equivalence on words that was originally introduced as strong Wilf equivalence by Kitaev et al. [Electron. J. Combin. 16(2)] 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 the same article by Kitaev et al. that the cardinality of each Wilf equivalence class is a power of $2$.
Highlights
In this work we investigate the notion of super-strong Wilf equivalence as given by J
We show that the answer to this is affirmative and, we give a full characterization of super-strong Wilf equivalence classes
In order to partition this set into super-strong Wilf equivalence classes, we define a labeling on the vertices of T n(u) that have two children, distinguishing between “good” ones which preserve symmetry, and “bad” ones which destroy symmetry
Summary
In this work we investigate the notion of super-strong Wilf equivalence as given by J. The intersection rule implies preservation of distances under super-strong Wilf equivalence This led us to define the notion of cross equivalence (see Section 4). In order to partition this set into super-strong Wilf equivalence classes, we define a labeling on the vertices of T n(u) that have two children, distinguishing between “good” ones which preserve symmetry (labeled 0), and “bad” ones which destroy symmetry (labeled 1). This labeling, which is in accordance to the sequence of differences ∆i(u−1) for i ∈ [2, n − 1], implies that the cardinality of each super-strong Wilf equivalence class is a power of 2
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