A Wilf Class Composed of 7 Symmetry Classes of Triples of 4-Letter Patterns

Abstract

We determine all 242 Wilf classes of triples of 4-letter 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 (agree in the first 16 terms) are in fact identical. The insertion encoding algorithm (INSENC) accounts for many of these and some others have been previously counted; in this paper, we find the generating function for each of the remaining 36 triples and it turns out to be algebraic in every 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 case and a bijection is used for another.

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

“…which implies (6). Now let b n denote the number of T -avoiding permutations of length n whose leftmost ascent is of the form a, n for some 2 ≤ a ≤ n − 1.…”

“…A Wilf class is said to be small or large depending on whether it contains one or more symmetry classes. The symmetry classes in large Wilf classes were enumerated in [6,7] (combined in [8]). The small Wilf classes that can be enumerated by the insertion encoding algorithm (INSENC) [18] are listed in Table 2 in the appendix.…”

On permutations avoiding 1324, 2143, and another 4-letter pattern

“…The contribution of the case π (1) > π (m) is given by xC(x)Q m−1 (x), where C(x) counts the permutations π (1) that avoid 132. So we can assume that π (m) has a letter greater than the smallest letter of π (1) . Since π avoids 2143, we see that π (j) = ∅ for all j = 2, 3, .…”

On permutations avoiding 1243, 2134, and another 4-letter pattern

“…, τ r . Here, we count the set S n (1243, 2134, τ ) for all 22 permutations τ ∈ S 4 \{1243, 2134} (for counting the set S n (T ) with T ⊆ S 4 , see [1,3,4,5,6,7,8]). The three involutions reverse, complement, invert on permutations generate a dihedral group that divides pattern sets into so-called symmetry classes.…”

“…• π (2) = ∅ and π (3) has a letter between i 1 and i 2 : Again, in this subcase, π can be expressed as…”