The Möbius Function of the Consecutive Pattern Poset

An occurrence of a consecutive permutation pattern p in a permutation π is a segment of consecutive letters of π whose values appear in the same order of size as the letters in p. The set of all permutations forms a poset with respect to such pattern containment. We compute the Möbius function of intervals in this poset. For most intervals our results give an immediate answer to the question. In the remaining cases, we give a polynomial time algorithm to compute the Möbius function. In particular, we show that the Möbius function only takes the values −1, 0 and 1. 1 Preliminaries and introduction For the poset of classical permutation patterns, the first results about its Möbius function were obtained in [SV]. Further results appear in [ST] and [BJJS]. The general problem in this case of classical patterns seems quite hard. In contrast, the poset of consecutive pattern containment has a much simpler structure. In this paper we compute the Möbius function of that poset. In most cases our results give an immediate answer. In the remaining cases, we give a polynomial time recursive algorithm to compute the ∗Steingŕımsson was supported by grant no. 090038012 from the Icelandic Research Fund. the electronic journal of combinatorics 18 (2011), #P146 1 Möbius function. In particular, we show that the Möbius function only takes the values −1, 0 and 1. An interesting result to note in connection to this is Björner’s paper [Bj] on the Möbius function of factor order. Although that poset is quite different from ours, there are interesting similarities. In particular, both deal with consecutive subwords and the possible values of the Möbius function are −1, 0 and 1 in both cases. Unless otherwise specified, all permutations in this paper are taken to be of the set [d] = {1, 2, . . . , d} for some positive integer d. We denote by Sd the set of all such permutations for a given d. An occurrence of a consecutive pattern σ = a1a2 . . . ak in a permutation τ = b1b2 . . . bn is a subsequence bi+1bi+2 . . . bi+k in τ , whose letters appear in the same order of size as the letters in σ. As an example, there are three occurrences of the consecutive pattern 231 in the permutation 563724891, namely 563, 372 and 891. On the other hand, the permutation 253641 avoids 231, since it contains no consecutive occurrence of that pattern. Consecutive permutation patterns are special cases of the generalized permutation patterns introduced in [BS], and they are not to be confused with the classical permutation patterns, whose occurrences in a permutation do not have to be contiguous. The enumerative properties of occurrences of various consecutive permutation patterns were first studied systematically in [EN], but these results will not concern us, as there seems to be no connection between them and the Möbius function studied here. The set of all permutations forms a poset P with respect to consecutive pattern containment. In other words, if σ ∈ Sk and τ ∈ Sn, then σ ≤ τ in P if σ occurs as a consecutive pattern in τ . We write σ < τ if σ ≤ τ and σ 6= τ . As usual in poset terminology, a permutation τ covers σ (and σ is covered by τ) if σ < τ and there is no permutation π such that σ < π < τ . Note that if τ covers σ then |τ | − |σ| = 1, where |π| is the length of π. The interval [x, y] in a poset P , where x and y are elements of P , is defined by [x, y] = {z ∈ P | x ≤ z ≤ y}. The rank of an interval [σ, τ ] in P is the difference |τ | − |σ|. The rank of an element π in [σ, τ ] is defined to be the rank of the interval [σ, π]. A filter in a poset P is a set S ⊆ P such that if x > y and y ∈ S, then x ∈ S. An ideal is a set S ⊆ P such that if x < y and y ∈ S, then x ∈ S. A principal filter is a filter with a single minimal element, and a principal ideal is an ideal with a single maximal element. In each case, the single minimal/maximal element is said to generate the filter/ideal. In the poset P consider the interval [σ, τ ]. Our aim is to compute μ(σ, τ), where μ is the Möbius function of the incidence algebra of P. The Möbius function is recursively defined by setting μ(x, x) = 1 for all x, and, if x 6= y,