Pattern-restricted permutations composed of 3-cycles

TL;DR

This paper characterizes and enumerates permutations made of 3-cycles that avoid specific patterns, providing explicit formulas and connections to well-known combinatorial sequences like Motzkin, Catalan, and Dyck paths.

Contribution

It offers a complete characterization and enumeration of pattern-avoiding permutations composed of 3-cycles for all six length-3 patterns, including explicit formulas and generating functions.

Findings

Number of 3-cycle permutations avoiding 231 pattern is 3^{n-1}.

Generating functions for pattern 132 avoidance relate to Motzkin and Catalan numbers.

Avoidance of 321 pattern involves weighted sums over Dyck paths.

Abstract

In this paper, we characterize and enumerate pattern-avoiding permutations composed of only 3-cycles. In particular, we answer the question for the six patterns of length 3. We find that the number of permutations composed of $n$ 3-cycles that avoid the pattern 231 (equivalently 312) is given by $3^{n-1}$, while the generating function for the number of those that avoid the pattern 132 (equivalently 213) is given by a formula involving the generating functions for the well-known Motzkin numbers and Catalan numbers. The number of permutations composed of $n$ 3-cycles that avoid the pattern 321 is characterized by a weighted sum involving statistics on Dyck paths of semilength~$n$.