Words Restricted by Patterns with at Most 2 Distinct Letters

We find generating functions for the number of words avoiding certain patterns or sets of patterns with at most 2 distinct letters and determine which of them are equally avoided. We also find exact numbers of words avoiding certain patterns and provide bijective proofs for the resulting formulae. Let [k] = {1, 2, . . . , k} be a (totally ordered) alphabet on k letters. We call the elements of [k] words. Consider two words, σ ∈ [k] and τ ∈ [`]. In other words, σ is an n-long k-ary word and τ is an m-long `-ary word. Assume additionally that τ contains all letters 1 through `. We say that σ contains an occurrence of τ , or simply that σ contains τ , if σ has a subsequence order-isomorphic to τ , i.e. if there exist 1 ≤ i1 < . . . < im ≤ n such that, for any relation φ ∈ {<, =, >} and indices 1 ≤ a, b ≤ m, σ(ia)φσ(ib) if and only if τ(a)φτ(b). In this situation, the word τ is called a pattern. If σ contains no occurrences of τ , we say that σ avoids τ . Up to now, most research on forbidden patterns dealt with cases where both σ and τ are permutations, i.e. have no repeated letters. Some papers (Albert et al. [AH], Burstein [B], Regev [R]) also dealt with cases where only τ is a permutation. In this paper, we consider some cases where forbidden patterns τ contain repeated letters. Just like [B], this paper is structured in the manner of Simion and Schmidt [SS], which was the first to systematically investigate forbidden patterns and sets of patterns. 1 Preliminaries Let [k](τ) denote the set of n-long k-ary words which avoid pattern τ . If T is a set of patterns, let [k](T ) denote the set of n-long k-ary words which simultaneously avoid all patterns in T, that is [k](T ) = ∩τ∈T [k] (τ). the electronic journal of combinatorics 9(2) (2002), #R3 1 For a given set of patterns T, let fT (n, k) be the number of T -avoiding words in [k] , i.e. fT (n, k) = |[k] (T )|. We denote the corresponding exponential generating function by FT (x; k); that is, FT (x; k) = ∑ n≥0 fT (n, k)x /n!. Further, we denote the ordinary generating function of FT (x; k) by FT (x, y); that is, FT (x, y) = ∑ k≥0 FT (x; k)y . The reason for our choices of generating functions is that k ≥ |[k](T )| ≥ n! ( k n ) for any set of patterns with repeated letters (since permutations without repeated letters avoid all such patterns). We also let GT (n; y) = ∑∞ k=0 fT (n, k)y , then FT (x, y) is the exponential generating function of GT (n; y). We say that two sets of patterns T1 and T2 belong to the same cardinality class, or Wilf class, or are Wilf-equivalent, if for all values of k and n, we have fT1(n, k) = fT2(n, k). It is easy to see that, for each τ , two maps give us patterns Wilf-equivalent to τ . One map, r : τ(i) 7→ τ(m+1−i), where τ is read right-to-left, is called reversal ; the other map, where τ is read upside down, c : τ(i) 7→ ` + 1 − τ(i), is called complement. For example, if ` = 3, m = 4, then r(1231) = 1321, c(1231) = 3213, r(c(1231)) = c(r(1231)) = 3123. Clearly, c ◦ r = r ◦ c and r = c = (c ◦ r) = id, so 〈r, c〉 is a group of symmetries of a rectangle. Therefore, we call {τ, r(τ), c(τ), r(c(τ))} the symmetry class of τ . Hence, to determine cardinality classes of patterns it is enough to consider only representatives of each symmetry class. 2 Two-letter patterns There are two symmetry classes here with representatives 11 and 12. Avoiding 11 simply means having no repeated letters, so f11(n, k) = (