Enumerating two permutation classes by the number of cycles

TL;DR

This paper provides a detailed enumeration of permutations within two specific classes based on their cycle counts, offering refined statistical insights into their structure.

Contribution

It introduces a novel enumeration approach for two permutation classes by cycle counts and refines this with additional permutation statistics.

Findings

Enumeration formulas for permutations by cycle counts

Refined statistical analysis of permutation classes

Enhanced understanding of permutation structure

Abstract

We enumerate permutations in the two permutation classes $\text{Av}_n(312, 4321)$ and $\text{Av}_n(321, 4123)$ by the number of cycles each permutation admits. We also refine this enumeration with respect to several statistics.