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Two permutations $\pi$ and $\tau$ are c-Wilf equivalent if, for each $n$, thenumber of permutations in $S_n$ avoiding $\pi$ as a consecutive pattern (i.e.,in adjacent positions) is the same as the number of those avoiding $\tau$. Inaddition, $\pi$ and $\tau$ are strongly c-Wilf equivalent if, for each $n$ and$k$, the number of permutations in $S_n$ containing $k$ occurrences of $\pi$ asa consecutive pattern is the same as for $\tau$. In this paper we introduce athird, more restrictive equivalence relation, defining $\pi$ and $\tau$ to besuper-strongly c-Wilf equivalent if the above condition holds for any set ofprescribed positions for the $k$ occurrences. We show that, when restricted tonon-overlapping permutations, these three equivalence relations coincide.  We also give a necessary condition for two permutations to be strongly c-Wilfequivalent. Specifically, we show that if $\pi,\tau$ in $S_m$ are stronglyc-Wilf equivalent, then $|\pi_m-\pi_1|=|\tau_m-\tau_1|$. In the special case ofnon-overlapping permutations $\pi$ and $\tau$, this proves a weaker version ofa conjecture of the second author stating that $\pi$ and $\tau$ are c-Wilfequivalent if and only if $\pi_1=\tau_1$ and $\pi_m=\tau_m$, up to trivialsymmetries. Finally, we strengthen a recent result of Nakamura andKhoroshkin-Shapiro giving sufficient conditions for strong c-Wilf equivalence.

Wilf equivalence relations for consecutive patterns