Enumerations for Permutations by Circular Peak Sets

TL;DR

This paper studies the enumeration of permutations based on their circular peak sets, providing explicit formulas, recurrence relations, and general formulas for various cases.

Contribution

It introduces explicit formulas and recurrence relations for counting permutations with specific circular peak sets, advancing combinatorial enumeration methods.

Findings

Derived explicit formulas for permutations with 0, 1, 2 circular peaks.

Established recurrence relations for the sequence $cp_n(S)$.

Provided a general formula for counting 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 $cp_n(S)$ 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 $cp_n(S)$. We also obtain some recurrence relations for the sequence $cp_n(S)$ and give the formula for $cp_n(S)$ in the general case.