Let $\pi \in \mathfrak{S}_m$ and $\sigma \in \mathfrak{S}_n$ be permutations. An occurrence of $\pi$ in $\sigma$ as a consecutive pattern is a subsequence $\sigma_i \sigma_{i+1} \cdots \sigma_{i+m-1}$ of $\sigma$ with the same order relations as $\pi$. We say that patterns $\pi, \tau \in \mathfrak{S}_m$ are strongly c-Wilf equivalent if for all $n$ and $k$, the number of permutations in $\mathfrak{S}_n$ with exactly $k$ occurrences of $\pi$ as a consecutive pattern is the same as for $\tau$. In 2018, Dwyer and Elizalde conjectured (generalizing a conjecture of Elizalde from 2012) that if $\pi, \tau \in \mathfrak{S}_m$ are strongly c-Wilf equivalent, then $(\tau_1, \tau_m)$ is equal to one of $(\pi_1, \pi_m)$, $(\pi_m, \pi_1)$, $(m+1 - \pi_1, m+1-\pi_m)$, or $(m+1 - \pi_m, m+1 - \pi_1)$. We prove this conjecture using the cluster method introduced by Goulden and Jackson in 1979, which Dwyer and Elizalde previously applied to prove that $|\pi_1 - \pi_m| = |\tau_1 - \tau_m|$. A consequenc 查看全文>>