Packing Patterns into Words

In this article we generalize packing density problems from permutations to patterns with repeated letters and generalized patterns. We are able to find the packing density for some classes of patterns and several other short patterns. A string 213322 contains three subsequences 233, 133, 122 each of which is orderisomorphic (or simply isomorphic) to the string 122, i.e. ordered in the same way as 122. In this situation we call the string 122 a pattern. Herb Wilf first proposed the systematic study of pattern containment in his 1992 address to the SIAM meeting on Discrete Mathematics. However, several earlier results on pattern containment exist, for example, those by Knuth [7] and Tarjan [11]. Most results on pattern containment actually deal with pattern avoidance, in other words, enumerate or consider properties of strings over a totally ordered alphabet which avoid a given pattern or set of patterns. There is considerably less research on other aspects of pattern containment, specifically, on packing patterns into strings over a totally ordered alphabet (but see [1, 3, 6, 8, 10]). In fact, all pattern packing except the one in [10] (later generalized in [1]) dealt with packing permutation patterns into permutations (i.e. strings without repeated letters). In this paper, we generalize the packing statistics and results to patterns over strings with repeated letters and relate them to the corresponding results on permutations. the electronic journal of combinatorics 10 (2003), #R00 1 1 Preliminaries Let [k] = {1, 2, . . . , k} be our canonical totally ordered alphabet on k letters, and consider the set [k] of n-letter words over [k]. We say that a pattern π ∈ [l] occurs in σ ∈ [k], or π hits σ, or that σ contains the pattern π, if there is a subsequence of σ order-isomorphic to π. Given a word σ ∈ [k] and a set of patterns Π ⊆ [l], let ν(Π, σ) be the total number of occurrences of patterns in Π (Π-patterns, for short) in σ. Obviously, the largest possible number of Π-occurrences in σ is ( n m ) , when each subsequence of length m of σ is an occurrence of a Π-pattern. Define μ(Π, k, n) = max{ ν(Π, σ) |σ ∈ [k]}, d(Π, σ) = ν(Π, σ) ( n m ) and δ(Π, k, n) = μ(Π, k, n) ( n m ) = max{ d(Π, σ) |σ ∈ [k]}, respectively, the maximum number of Π-patterns in a word in [k], the probability that a subsequence of σ of length m is an occurrence of a Π-pattern, and the maximum such probability over words in [k]. We want to consider the asymptotic behavior of δ(Π, k, n) as n→∞ and k →∞. Proposition 1.1 If n > m, then δ(Π, k, n) ≤ δ(Π, k, n−1) and δ(Π, k, n) ≥ δ(Π, k−1, n). Proof. The proof of Proposition 1.1 in [1] also applies to the first inequality in our proposition, since possible repetition of letters is irrelevant here. To see that the second inequality is true, note that increasing k, i.e. allowing more letters in our alphabet, can only increase μ(Π, k, n), and hence δ(Π, k, n). 2 The greatest possible number of distinct letters in a word σ of length n is n, which implies that μ(Π, k, n) = μ(Π, n, n) for k ≥ n, and hence, δ(Π, k, n) = δ(Π, n, n) for k ≥ n. Therefore, δ(Π, n, n) = lim k→∞ δ(Π, k, n). We also have δ(Π, n, n) = δ(Π, n+1, n) ≥ δ(Π, n+1, n+1), so δ(Π, n, n) is non-increasing and nonnegative, and there exists δ(Π) = lim n→∞ δ(Π, n, n) = lim n→∞ lim k→∞ δ(Π, k, n). We call δ(Π) the packing density of Π. Obviously, there are two double limits. Since 0 ≤ δ(Π, k, n) ≤ 1, it immediately follows that there exists δ(Π, k) = lim n→∞ δ(Π, k, n) ∈ [0, 1] and that {δ(Π, k) | k ∈ N} is nondecreasing as k →∞. Hence, there exists δ′(Π) = lim k→∞ δ(Π, k) = lim k→∞ lim n→∞ δ(Π, k, n). the electronic journal of combinatorics 10 (2003), #R00 2 It is easy to see that δ′(Π) ≤ δ(Π). Naturally, one wishes to determine when δ′(Π) = δ(Π). In this paper, we will provide a sufficient condition for this equality. The set [k] is finite, so for each k and n, there is a string σ(Π, k, n) ∈ [k] such that d(Π, σ(Π, k, n)) = δ(Π, k, n). To find δ(Π), we will need to find δ(Π, k, n), hence maximal Π-containing permutations σ(Π, k, n) are of interest to us, especially, their asymptotic shape as n→∞ and k →∞. Example 1.2 Let Π = {cm}, where cm is a constant string of m 1’s. Then, clearly, σ(Π, k, n) = cn and d(cm, cn) = 1 for n ≥ m, so δ(cm, k, n) = 1 for n ≥ m, and hence δ(cm) = δ(cm) = 1 for any m ≥ 1. Example 1.3 Let Π = {idm}, where idm is the identity permutation of Sm. Then σ(idm, n, n) = idn, so d(idm, idn) = 1, δ(idm, n, n) = 1 and δ(idm) = 1. Determining δ(idm) is a bit harder. It is easy to see that σ(idm, k, n) must be a nondecreasing string of digits in [k]. Let ni be the number of digits i in σ(idm, k, n), then μ(idm, k, n) = ν(idm, σ(idm, k, n)) = n1n2 . . . nk and n1 + n2 + · · ·+ nk = n. To maximize the above product we need n1 = n2 = · · · = nk = nk . (More exactly, [8] shows that we should choose for ni’s to be such integers that |ni − nk | < 1 and |n1 + · · · + nr − rn k | < 1 for each r = 1, 2, . . . , k.) It follows that δ(idm, k, n) ∼ ( k m ) ( n k )m ( n m ) (where an ∼ bn means limn→∞ an/bn = 1), so δ(idm, k) = ( k m ) m! km , and thus δ(idm) = 1 as expected. Packing density was initially defined for patterns in permutations. Therefore, we must show that the packing density on permutations agrees with the packing density on words. Theorem 1.4 Let Π ⊆ Sm be a set of permutation patterns, then δ(Π) = lim n→∞ max{ ν(Π, σ) |σ ∈ Sn} ( n m ) , i.e. the packing density of Π on words is equal to that on permutations. Proof. It is enough to prove that μ(Π, n, n) = max{ ν(Π, σ) |σ ∈ Sn}, in other words, that there is a permutation in Sn among the maximal Π-containing words in [n]. Consider any maximal Π-containing word σ ∈ [n]. Let ni be the multiplicity of the letter i in σ. Let ij denote the jth occurrence of the letter i, and consider the map f : [n] → Sn induced by the map ij 7→ ∑i r=1 nr − j + 1. Since all letters of each pattern in Π are distinct, Π occurs in f(σ) at least at the same positions Π occurs in σ, so ν(Π, f(σ)) ≥ ν(Π, σ). The rest is easy. 2 the electronic journal of combinatorics 10 (2003), #R00 3 Apart from computing packing densities of patterns, we would also like to determine which patterns have equal packing densities, which ones are asymptotically more packable than others, etc. For example, it is easy to see that the packing density is invariant under the usual symmetry operations on [l]: reversal r : τ(i)→ τ(m− i+ 1) and complement c : τ(i)→ l − τ(i) + 1, (packing density is also invariant under inverse i : τ → τ−1 when packing permutations into permutations). The operations r and c generateD2, while r, c, i generate D4. Patterns which can be obtained from each other by a sequence of symmetry operations are said to belong to the same symmetry class. Example 1.5 The symmetry class representatives of patterns in [3] are 111, 112, 121, 123 and 132. We know that δ(111) = 1 = δ(123). Galvin, Kleitmann and Stromquist (independently, unpublished, see chronology in [8]) showed that δ(132) = 2 √ 3− 3 ≈ 0.4641. Thus, we only need to determine the packing densities of 112 and 121 to completely classify patterns of length 3. Price [8] extended Stromquist’s results [10] to packing a single pattern π = 1m(m − 1) . . . 2 and handled other single patterns such as 2143. Since we will also be concerned mostly with singleton sets of patterns Π = {π}, we will write δ(π) for δ({π}), etc. Price’s results deal with patterns of specific type, the so-called layered patterns. Definition 1.6 A layered pattern is a strictly increasing sequence of strictly decreasing substrings. These substrings are called the layers of σ. Notation 1.7 It easy to see that a layered pattern is uniquely determined by the sequence of its layer lengths, hence we may denote it by such sequence, e.g. 3̂21 5̂4 9̂876 = [3, 2, 4], 1̂2̂3̂ = [1, 1, 1], 1̂3̂2 = [1, 2], 2̂13̂ = [2, 1], 3̂21 = [3] are layered, with layers denoted by hats, while 312, 231 are non-layered. In fact, note that the union of symmetry classes of layered patterns consists of exactly the permutations avoiding patterns in the symmetry classes of 1342, 1423, 2413. In [10], Stromquist proved a theorem (later generalized in [1]) on packing layered patterns into permutations. The inductive proof of this theorem defines a permutation (or a poset) π to be layered on top (or LOT ) if any of its maximal elements is greater than any non-maximal element. The set of these maximal elements is called the final layer of π (even if π is not necessarily layered). Proposition 1.8 Let Π be a multiset of LOT permutations (not necessarily all distinct or of equal length). Then there is an LOT permutation σ∗ which maximizes the expression