Wilf equivalence relations for consecutive patterns

We also give a necessary condition for two permutations to be strongly c-Wilf equivalent. Specifically, we show that if π,τ∈Sm are strongly c-Wilf equivalent, then |πm−π1|=|τm−τ1|. In the special case of non-overlapping permutations π and τ, this proves a weaker version of a conjecture of the second author stating that π and τ are c-Wilf equivalent if and only if π1=τ1 and πm=τm, up to trivial symmetries. Finally, we strengthen a recent result of Nakamura and Khoroshkin-Shapiro giving sufficient conditions for strong c-Wilf equivalence.