Enumerations for Permutations by Circular Peak Sets

Abstract

The circular peak set of a permutation $\sigma$ is the set $\{\sigma (i)\mid \sigma (i-1)<\sigma (i)>\sigma (i+1)\}$. In this paper, we focus on the enumeration problems for permutations by circular peak sets. Let $c{p}_n(S)$ denote the number of the permutations of order $n$ which have the circular peak set $S$. For the case with $|S|=0,1,2$, we derive the explicit formulas for $c{p}_n(S)$. We also obtain some recurrence relations for the sequence $c{p}_n(S)$ and give the formula for $c{p}_n(S)$ in the general case.