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Wilf classification of triples of 4-letter patterns II

Wilf classification of triples of 4-letter patterns II

This is the second of two papers in which we determine all 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces to showing that counting sequences that appear to be the same (i.e., agree in the first 16 terms) are in fact identical. This amounts to counting avoiders for 107 representative triples. The insertion encoding algorithm (INSENC) applies to many of them and some others have been previously counted. There remain 36 triples and the first paper dealt with the first 18. In this paper, we find the generating function for the last 18 triples which turns out to be algebraic in each case. Our methods are both combinatorial and analytic, including decompositions by left-right maxima and by initial letters. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence that succumbs to the kernel method. A particularly nice so-called cell decomposition is used in one of the cases (Case 238)