Enumeration of permutations avoiding a triple of 4-letter patterns is all done

TL;DR

This paper completes the enumeration of permutations avoiding specific triples of 4-letter patterns by providing algebraic generating functions for all remaining cases, resolving previous gaps in the classification.

Contribution

It determines algebraic generating functions for 13 previously unresolved pattern triples, completing the enumeration of all 317 symmetry classes.

Findings

Generated algebraic functions for 13 pattern triples

Confirmed the algebraic nature of all but one class

Resolved the enumeration problem for all symmetry classes

Abstract

This paper completes a project to enumerate permutations avoiding a triple T of 4-letter patterns, in the sense of classical pattern avoidance, for every T. There are 317 symmetry classes of such triples T and previous papers have enumerated avoiders for all but 14 of them. One of these 14 is conjectured not to have an algebraic generating function. Here, we find the generating function for each of the remaining 13, and it is algebraic in each case.