Cycle up-down permutations

TL;DR

This paper introduces cycle-up-down permutations, proves they are enumerated by Euler numbers, and analyzes various permutation statistics through bijective and analytical methods.

Contribution

It provides a novel characterization and enumeration of cycle-up-down permutations, linking them to Euler numbers and exploring their statistical properties.

Findings

Cycle-up-down permutations are counted by Euler numbers.

Distribution of cycle and element statistics is characterized.

Connections to up-down permutations and generalizations are established.

Abstract

A permutation is defined to be cycle-up-down if it is a product of cycles that, when written starting with their smallest element, have an up-down pattern. We prove bijectively and analytically that these permutations are enumerated by the Euler numbers, and we study the distribution of some statistics on them, as well as on up-down permutations, on all permutations, and on a generalization of cycle-up-down permutations. The statistics include the number of cycles of even and odd length, the number of left-to-right minima, and the number of extreme elements.