Enumeration of permutations by number of cyclic occurrence of peaks and valleys

TL;DR

This paper studies how permutations can be counted based on the cyclic occurrences of peaks and valleys, establishing recurrence relations, generating functions, and connections to Pell numbers and alternating runs.

Contribution

It introduces new recurrence relations and links cyclic peaks and valleys to well-known sequences like Pell numbers and alternating runs.

Findings

Derived recurrence relations for permutations with fixed cyclic peaks and valleys

Established generating functions for related permutation statistics

Connected cyclic valleys to Pell numbers and cyclic peaks to alternating runs

Abstract

In this paper, we focus on the enumeration of permutations by number of cyclic occurrence of peaks and valleys. We find several recurrence relations involving the number of permutations with a prescribed number of cyclic peaks, cyclic valleys, fixed points and cycles. Several associated permutation statistics and the corresponding generating functions are also studied. In particular, we establish a connection between cyclic valleys and Pell numbers as well as cyclic peaks and alternating runs.