Wilf-Equivalence on $k$-ary Words, Compositions, and Parking Functions

Abstract

In this paper, we study pattern-avoidance in the set of words over the alphabet $[k]$. We say that a word $w\in[k]^n$ contains a pattern $\tau\in[\ell]^m$, if $w$ contains a subsequence order-isomorphic to $\tau$. This notion generalizes pattern-avoidance in permutations. We determine all the Wilf-equivalence classes of word patterns of length at most six. We also consider analogous problems within the set of integer compositions and the set of parking functions, which may both be regarded as special types of words, and which contain all permutations. In both these restricted settings, we determine the equivalence classes of all patterns of length at most five. As it turns out, the full classification of these short patterns can be obtained with only a few general bijective arguments, which are applicable to patterns of arbitrary size.

Wilf-collapse in permutation classes having two basis elements of size three

“…In other words, some subword of B of the same length as A dominates A term by term. This is a somewhat different pattern containment for compositions to that considered in [8] or [11] and so the results below do not apply to those contexts.…”

On (shape-)Wilf-equivalence for words

“…Figure 3. A 0-1-filling of shape (10,10,10,7,4,4) More generally, given a set Ω of words, we write W (a 1 ,a 2 ,... ) λ…”

“…Remark. By going through the proof of Proposition 6 in [7], one sees that the validity of Conjecture 11 would imply that…”

“…This construction is illustrated in Figure 4. The left of the figure shows a filling in W (2,2,3,1,1,1) (10,10,10,7,4,4) (231, 221). The filling on the right shows the result of the above described conversion.…”

“…In the figure, the separations between the original rows are indicated by thick lines, whereas the newly created rows are separated by thin lines. The resulting filling is in W (1,1,...,1) λ 1 (231), where λ 1 = (10, 10, 10, 10, 10, 10, 10,7,4,4). We now apply the bijection α to R 1 , to obtain a 312-avoiding full rook placement R 2 .…”