On the sub-permutations of pattern avoiding permutations

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  • Discrete Mathematics 337 (2014) 127141

    Contents lists available at ScienceDirect

    Discrete Mathematics

    journal homepage: www.elsevier.com/locate/disc

    On the sub-permutations of pattern avoiding permutationsFilippo Disanto , Thomas WieheUniversitt zu Kln, Institut fr Genetik, Zlpicher Strae 47a, 50674 Kln, Germany

    a r t i c l e i n f o

    Article history:Received 6 November 2012Received in revised form 20 March 2014Accepted 29 June 2014Available online 28 July 2014

    Keywords:PermutationSub-permutationPattern avoidanceBinary increasing treesPattern-search

    a b s t r a c t

    There is a deep connection between permutations and trees. Certain sub-structures ofpermutations, called sub-permutations, bijectively map to sub-trees of binary increasingtrees. This opens a powerful tool set to study enumerative and probabilistic properties ofsub-permutations and to investigate the relationships between local and global featuresusing the concept of pattern avoidance.

    First, given a pattern , we study how the avoidance of in a permutation affectsthe presence of other patterns in the sub-permutations of . More precisely, consideringpatterns of length 3, we solve instances of the following problem: given a class ofpermutations K and a pattern , we ask for the number of permutations Avn()whose sub-permutations in K satisfy certain additional constraints on their size.

    Second, we study the probability for a generic pattern to be contained in a randompermutation of size n without being present in the sub-permutations of generated bythe entry 1 k n. These theoretical results can be useful to define efficient randomizedpattern-search procedures based on classical algorithms of pattern-recognition, while thegeneral problem of pattern-search is NP-complete.

    2014 Elsevier B.V. All rights reserved.

    1. Introduction

    Characterizing a class of objects by means of its sub-structures is a general theme in mathematics. Using the conceptof pattern avoidance, we investigate sub-structures of permutations, called sub-permutations, and demonstrate bijectiverelations and enumerative and probabilistic results.

    Sub-permutations, as defined in Section 2, correspond to the classical concept of sub-trees via a well known [11,23]bijection which maps the set of permutations of size n onto the set of binary increasing trees (BITs) with n nodes.

    The study of patterns in sub-permutations of un-constrained random permutations has been initiated [8] by Flajoletet al. Using the equivalent sub-tree terminology for BITs, they focus on the number of times a fixed permutation occurs as asub-permutation of a larger random permutation.

    There is a long standing interest in studying discrete structures constrained by pattern conditions. The concept of patternavoidance was introduced by MacMahon [16] and then elaborated by Knuth [14] and by Simion and Schmidt [21]. Lately,several combinatorial structures trees [6,8,10,19], lattice paths [4,20] and compositions of integers [13] have beenanalysed using patterns constraints.

    Here, we expand these concepts to sub-permutations and, by means of equivalence, to sub-trees of BITs, whichare constrained by pattern avoiding conditions. This new point of view is introduced with the general aim of betterunderstanding the relationships between local and global features of pattern avoidance.

    Correspondence to: Stanford University, Department of Biology, 371 Serra Mall, Stanford, CA 94305, USA.E-mail addresses: fdisanto@uni-koeln.de, fdisanto@stanford.edu (F. Disanto), twiehe@uni-koeln.de (T. Wiehe).

    http://dx.doi.org/10.1016/j.disc.2014.06.0280012-365X/ 2014 Elsevier B.V. All rights reserved.

    http://dx.doi.org/10.1016/j.disc.2014.06.028http://www.elsevier.com/locate/dischttp://www.elsevier.com/locate/dischttp://crossmark.crossref.org/dialog/?doi=10.1016/j.disc.2014.06.028&domain=pdfmailto:fdisanto@uni-koeln.demailto:fdisanto@stanford.edumailto:twiehe@uni-koeln.dehttp://dx.doi.org/10.1016/j.disc.2014.06.028

  • 128 F. Disanto, T. Wiehe / Discrete Mathematics 337 (2014) 127141

    a

    b

    Fig. 1. (a) The permutation = 45312687; (b) the tree associated with by .

    In Section 3 we concentrate on patterns of length three. We derive several enumerative and bijective properties for sub-permutations, relating themwith other well-studied combinatorial structures such as planar binary trees and lattice paths.These properties are most easily formulated in the light of the following general problem.

    Problem. Given a class of permutations K , we ask for the number of permutations Avn() whose sub-permutationsin K satisfy certain additional constraints on their size.

    In particular, we consider = 312 and = 123. These two patterns can be taken as representative of the two clustersof patterns {312, 213, 132, 231} and {123, 321} which are not equivalent with respect to the standard operators of mirrorimage, complement and inverse. In Section 3 we consider four possible instances of class K . For = 312 we considerK = Av(213) and K equal to the set of odd alternating permutations. In particular we prove that, for n , taking atrandom a permutation in Avn(312) the expected value of the size of the biggest sub-permutation of belonging toAv(213) grows with log2(n). For = 123 we consider K = Av(21) and K = Av(12). For the latter, we show that thenumber of permutations of Avn(123), whose biggest non-trivial decreasing sub-permutation has size bounded by a fixednumber j, is related to the number of Dyck paths of size n avoiding the pattern U j+2D, where U (resp. D) stand for up (resp.down) step in the path.

    In Section 4we introduce the concept of pattern avoidance in termsof sub-permutations.We find theprobability to detecta pattern 213 in a permutation by looking just at the sub-permutation generated by the entry k = 2. Then, we generalize ourresults considering an arbitrary pattern and a general value of the parameter k and we show that our theoretical resultsare in agreement with data generated by Monte Carlo random experiments.

    We expect that this new kind of pattern related problems will draw a number of practical applications. For instance, forcomputational tasks [1,5], it can be important to know when global properties of a permutation can be predicted, if onlylocal properties are known or accessible. The kind of questions formulated here in the context of permutations and BITs aregeneric for all combinatorial objects for which a notion of sub-structure is defined.

    2. Preliminaries

    The set of permutations of size n is denoted by Sn and S =

    n Sn. We assume the reader to be familiar with the classicalconcept of pattern avoidance in permutations. The subset of Sn made of those permutations avoiding the pattern is usuallydenoted by Avn( ) and Av( ) =

    n(Avn( )).

    Let = 12 . . . n be a permutation. For a given entry i we define s (i) as the biggest sub-string1 of which con-tains i and whose entries are greater than or equal to i. Furthermore let g (i) be the permutation obtained rescalings (i). We call g (i) the sub-permutation of generated by i. The set of sub-permutations (g (i))i=1...n is denotedby G . Note that not all sub-strings of are (once rescaled) sub-permutations of . It is crucial that only those maxi-mal sub-strings are extracted from whose elements share the property of being greater than or equal to a given en-try of the sub-string. As an example consider the permutation which is depicted in Fig. 1(a). In this case G is made ofg (5) = g (8) = 1, g (4) = 12, g (7) = 21, g (3) = 231, g (6) = 132, g (2) = 1243 and g (1) = 45312687.

    We also observe that there is a strong correspondence between sub-permutations as defined above and the so-called x-factorizations of a permutation, see [12] and related references. Indeed, following the notation of [12], the sub-permutationg (i) is obtained by extracting together from the permutation the entry i and the and parts of the i-factorizationof .

    The concept of sub-permutations is related to the one of sub-trees. To illustrate this correspondence we make use of awell known bijection [11,23] between the set Sn and the set of binary increasing trees of size n, denoted by Tn. We recallthat a planar rooted tree t , having n nodes, belongs to Tn when:

    1 Sub-strings are sub-sequences of consecutive elements of .

  • F. Disanto, T. Wiehe / Discrete Mathematics 337 (2014) 127141 129

    each node has outdegree 0, 1 or 2. Nodes of outdegree 0 are called leaves; each node (except for the root) can be left or right oriented with respect to its direct ancestor (see Fig. 1(b)); each node is labelled (bijectively) with a number in {1, . . . , n} in such a way going from the root of t to any leaf of t wefind an increasing sequence of numbers.

    The bijection : Tn Sn is given by the following procedure:(i) given a tree t each leaf of t collapses into its direct ancestor whose label is then modified receiving on the left the label

    of the left child (if any) and on the right the label of the right child (if any). We obtain in this way a new tree whosenodes are labelled with sequences of numbers;

    (ii) starting from the obtained tree go to step (i).

    The algorithm ends when the tree t is reduced to a single node whose label is then a permutation (t) of size n. For anexample see Fig. 1.

    The link between sub-permutations and sub-trees is expressed by the following proposition.

    Proposition 1. Given a permutation = (t), let ti be the (re-scaled) sub-tree of t generated by the node i, then the sub-permutation g (i) is equal to (ti).

    By Proposition 1, we know that the size of the sub-permutation g (i) is given by the number of nodes descending fromnode i in the tree t = 1(). In [15] the authors study statistics related to the number of descendants in subtrees of BITs.In particular, for a random permutation of size n expectation and variance of the size of the sub-permutation generatedby entry k are

    E(|g (k)|) =2n k + 1

    k + 1and (1)

    Var(|g (k)|) =2(n + 1)(k 1)(n k)

    (k + 1)2(k + 2). (2)

    More generally, for a random permutation of size n, the size of the sub-permutation generated by k is equal to m withprobability

    Prob(|g (k)| = m) =km

    nm1k2

    n

    n1k1

    . (3)3. On the sub-permutations of Av() belonging to K

    In this section we determine the number of permutations in Avn() whose sub-permutations in K satisfy certain size-constraints. We consider several instances of and K and find new enumerative results depending on the constraintswe require each time. Some of them are connected to already known combinatorial problems. We will study the patterns = 312 and = 123. It is well known that both the classes Av(312) and Av(123) are enumerated by Catalan numberswhose first terms are

    c0 = 1, c1 = 1, c2 = 2, c3 = 5, c4 = 14, c5 = 42, c6 = 132, c7 = 429.

    The associated ordinary generating function is [9]

    C(x) =n0

    cnxn =1

    1 4x2x

    , (4)

    which has a closed form for its n-th coefficient given by

    cn =1

    n + 1

    2nn

    .

    Furthermore, it is well known that the asymptotic behaviour for cn is

    cn 4n

    n3

    .

    3.1. = 312 and binary rooted planar trees

    We denote by Bn the set of binary rooted planar trees with n internal nodes, where each internal node has outdegreetwo. It is well known that one can bijectively map the set Bn onto the set Avn(312). In particular we use a bijection : Bn Avn(312) which works similarly to the mapping of Section 2, as follows.

  • 130 F. Disanto, T. Wiehe / Discrete Mathematics 337 (2014) 127141

    Fig. 2. The mapping .

    Take t Bn and visit its nodes according to the pre-order traversal labelling each node of outdegree two in increasingorder starting with the label 1 for the root. After this first step one has a tree labelled with integers at its nodes of outdegreetwo. Each leaf now collapses to its direct ancestorwhich takes a new label receiving on the left (resp. right) the label of its left(resp. right) child. We go on collapsing leaves until we achieve a treemade of one nodewhich is labelled with a permutationof size n. See Fig. 2 for an instance of this mapping.

    3.1.1. K = Av(213) and caterpillar sub-treesLet K = Av(213). A tree t in Bn is a caterpillar of size n, if each node of t is a leaf or has at least one leaf as a direct

    descendant. Through the bijection we see that, given a permutation Av(312), the parameter (K), that is the sizeof the biggest sub-permutation of belonging to Av(213), corresponds to the size (number of internal nodes) of the biggestsub-tree of 1() which is a caterpillar.

    LetPj = Pj(x) =

    n0

    vj,nxn

    be the ordinary generating function counting those permutations in Avn() having (K) j. We observe that Pj mustthen satisfy the equation

    Pj = 1 + xPj2 2jxj+1. (5)Indeed, a tree t whose biggest caterpillar sub-tree is of size at most j, is either a leaf or it is built by appending to the root

    of t two trees t1 and t2 of the same class. We must exclude the case in which one of the two, t1 or t2, has size 0, i.e., is a leaf,and the other one is a caterpillar of size j. Since there are exactly 2j1 caterpillars of size j the previous formula follows.

    From (5) we obtain

    Pj(x) =1

    1 4x + 2j+2xj+2

    2x(6)

    which gives an expansion, for j = 1, with the following coefficients1, 1, 0, 1, 2, 6, 16, 45, 126, 358, 1024

    which correspond to sequence A025266 of [22]. As stated there, they also count classes of Motzkin paths satisfying someconstraints.

    The asymptotic behaviour of the coefficients of Pj follows by standard analytic methods [9]. When n becomes large thecoefficients grow like

    vj,n 14

    4j (j + 2)2j+2

    j+2j

    n3

    1j

    n+1, (7)

    where j is the smallest positive solution of

    1 4x + 2j+2xj+2 = 0.Starting from the following inequality

    14

    < j 2 < 3 > < n . It is well known that the number of strictly binary trees of size 2m + 1 is cm.

    The ordinary generating function Lj = Lj(x), with j = 2m + 1, counts those trees in Bn having at least one strictlybinary sub-tree of size j. Equivalently, it can be seen as the function counting the permutations in Avn() with at least onealternating sub-permutation of size j. Lj must then satisfy

    Lj = cmxj + xLj2 + 2xLj(C Lj), (10)

    where C = C(x) is the generating function of Catalan numbers as in (4).Indeed a tree counted by Lj is either a strictly binary tree of size j (first summand) or it is built by appending to the root

    two trees, where at least one of them contains a strictly binary sub-tree of size j (second and third summand).By solving (10) we obtain

    Lj(x) =

    1 4x + 4cmxj+1

    1 4x

    2x. (11)

    From (11), we can also determine the number of those permutations avoiding the pattern 312 and without (odd)alternating sub-permutation of size j. This is given by

    C(x) Lj(x) =1

    1 4x + 4cmxj+1

    2x. (12)

    It is now possible to compare results (9) and (12) for a fixed j = 2m + 1 when m is large. In other words, we want tomeasure how the avoidance of the pattern = 312 affects the presence of the pattern 213 in sub-permutations (from(9)) in comparison with the avoidance of alternating sub-permutations (from (12)). By means of the bijection with treesdescribed before, this gives a comparison be...