On a pattern avoidance condition for the wreath product of cyclic groups with symmetric groups

TL;DR

This paper extends pattern avoidance studies in wreath products of cyclic and symmetric groups to non-consecutive cases, providing new enumerative results and a novel interpretation of Catalan numbers through pattern bi-avoidance.

Contribution

It introduces a non-consecutive pattern matching condition in wreath products and offers a new combinatorial interpretation of Catalan numbers via pattern bi-avoidance.

Findings

Enumeration results for longer patterns

A bijective interpretation of Catalan numbers

Extension of pattern avoidance to non-consecutive cases

Abstract

In this paper, we extend to a non-consecutive case, the study of the pattern matching condition on the wreath product of the cyclic group and the symmetric group initiated by the authors in a previous paper. The main focus of our paper is (colored) patterns of length 2, although a number of enumerative results for longer patterns are also presented. A new non-trivial bijective interpretation for the Catalan numbers is found, in terms of simultaneously bi-avoiding two patterns in a wreath product.