Cyclically Consecutive Permutation Avoidance

Abstract

We give an explicit formula for the number of permutations that cyclically avoid a consecutive pattern in terms of the spectrum of the associated operator of the consecutive pattern. As an example, the number of cyclically consecutive 123-avoiding permutations in ${\mathfrak S}_{n}$ is given by $n!$ times the convergent series $\sum_{k=-\infty}^{\infty} (\frac{\sqrt{3}}{2\pi(k+1/3)})^{n}$ for $n \geq 2$.