Enumerating circular permutations avoiding the vincular pattern 23 4 1

TL;DR

This paper derives an explicit generating function for counting circular permutations avoiding a specific vincular pattern, solving an open problem and completing the enumeration of such patterns of length four.

Contribution

It provides the first explicit formula for the generating function counting circular permutations avoiding the pattern 23 4 1, addressing an open problem in combinatorics.

Findings

Explicit formula for the generating function is obtained.

Enumeration of circular permutations avoiding any single vincular pattern of length four is completed.

Methods include auxiliary arrays, functional equations, kernel method, and iteration.

Abstract

In this paper, we find an explicit formula for the generating function that counts the circular permutations of length n avoiding the pattern 23 4 1 whose enumeration was raised as an open problem by Rupert Li. This then completes in all cases the enumeration of circular permutations that avoid a single vincular pattern of length four containing one vinculum. To establish our results, we introduce three auxiliary arrays which when taken together refine the cardinality of the class of permutations in question. Rewriting the recurrences of these arrays in terms of generating functions leads to functional equations which are solved by various means including the kernel method and iteration.