We give a necessary condition for two permutations to be strongly
c-Wilf equivalent. Specifically, we show that if
π,τ in **S**_{m}
are strongly c-Wilf equivalent, then
|π_{m}-π_{1}|
= |τ_{m}-τ_{1}|. 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
π_{1} = τ_{1} and
π_{m} = τ_{m}, up to trivial
symmetries. Additionally, we show that for non-overlapping
permutations, c-Wilf equivalence coincides with super-strong c-Wilf
equivalence, and we strengthen a recent result of Nakamura and
Khoroshkin-Shapiro giving sufficient conditions for strong c-Wilf
equivalence.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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