Enumeration of small Wilf classes avoiding 1342 and two other $4$-letter patterns

TL;DR

This paper enumerates permutation classes avoiding specific 4-letter patterns, focusing on those containing 1342, and provides explicit algebraic generating functions for most classes.

Contribution

It completes the enumeration of small Wilf classes containing 1342, introducing 40 new cases with explicit algebraic generating functions.

Findings

Most classes have rational or degree 2 algebraic generating functions.

One class has an algebraic generating function of degree 3.

The methods involve detailed analysis of permutation maxima and initial letters.

Abstract

This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is shown to be algebraic of degree 3. We use standard methods, usually involving detailed consideration of the left to right maxima, and sometimes the initial letters, to obtain an algebraic or functional equation for the generating function.