Avoiding patterns of length three in compositions and multiset permutations

Heubach, Silvia; Mansour, Toufik

doi:10.1016/j.aam.2005.06.005

Cited by 22 publications

(18 citation statements)

References 3 publications

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“…In the following table we compare the values of the ratio of Proposition 3 with the asymptotic ratio (14) for m = 5 and different values of n. n 50 500 1000 5000 10000 ratio (13) 0.986 0.887 0.789 0.308 0.095 k m = 1 in (14) 0.988 0.889 0.791 0.310 0.096…”

Section: From These Results Followsmentioning

confidence: 99%

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On the sub-permutations of pattern avoiding permutations

Disanto

Wiehe

2014

Discrete Mathematics

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There is a deep connection between permutations and trees. Certain sub-structures of permutations, called sub-permutations, bijectively map to sub-trees of binary increasing trees. This opens a powerful tool set to study enumerative and probabilistic properties of sub-permutations and to investigate the relationships between 'local' and 'global' features using the concept of pattern avoidance. First, given a pattern mu, we study how the avoidance of mu in a permutation pi affects the presence of other patterns in the sub-permutations of pi. More precisely, considering patterns of length 3, we solve instances of the following problem: given a class of permutations K and a pattern mu, we ask for the number of permutations pi is an element of Av(n)(mu) 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 random permutation pi of size n without being present in the sub-permutations of pi generated by the entry 1 <= k <= n. These theoretical results can be useful to define efficient randomized pattern-search procedures based on classical algorithms of pattern-recognition, while the general problem of pattern-search is NP-complete. (C) 2014 Elsevier B.V. All rights reserved

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Section: From These Results Followsmentioning

confidence: 99%

“…The concept of pattern avoidance was introduced by MacMahon [16] and then elaborated by Knuth in [14] and in [21] by Simion and Schmidt. Lately, several combinatorial structures -trees [6,8,10,19], lattice paths [4,20] and compositions of integers [13] -have been analyzed using patterns constraints.…”

Section: Introductionmentioning

confidence: 99%

On the sub-permutations of pattern avoiding permutations

Disanto

Wiehe

2014

Discrete Mathematics

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“…Since linear extensions are a central object of study in order theory, it is natural to look at avoidance of more complicated patterns than 21 in poset permutations. The field of pattern avoidance has blossomed in the past two decades, and in particular has expanded to include patterns in structures other than S n such as words [2] [6], compositions and multiset permutations [3] [1] [19] [10], set partitions [12] [17], ordered set partitions [9], matchings [4], et cetera. In [11], Kitaev studies classical permutation avoidance of patterns with incomparable elements.…”

Section: Motivationmentioning

confidence: 99%

Pattern Avoidance in Poset Permutations

Hopkins

Weiler²

2015

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Abstract. We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on P that avoid the pattern π is denoted AvP (π). We extend a proof of Simion and Schmidt to show that AvP (132) ≤ AvP (123) for any poset P , and we exactly classify the posets for which equality holds. MotivationAn inversion of a permutation σ ∈ S n is a pair of entries a < b ∈ {1, . . . , n} such that b is to the left of a in σ. As Bóna [5] explains, classical pattern containment is "a far-fetching generalization of [inversions of permutations] from pairs of entries to k-tuples of entries." A permutation σ ∈ S n is said to contain a pattern π ∈ S k if some subsequence of σ of length k is orderisomorphic to π. Otherwise, we say σ avoids π. Thus an inversion in σ is just a 21 pattern that it contains.We propose a similar generalization to permutations on posets. A total ordering of the elements of some poset P that respects its partial order is called a linear extension of P . It is possible also to think of a linear extension of P as a permutation of the elements of P that has no inversions. While the only classical permutation in S n that has no inversions is 12 . . . n, in general there may be many linear extensions of P , and counting the number of such extensions is a difficult problem. Since linear extensions are a central object of study in order theory, it is natural to look at avoidance of more complicated patterns than 21 in poset permutations. The field of pattern avoidance has blossomed in the past two decades, and in particular has expanded to include patterns in structures other than S n such as words , Kitaev studies classical permutation avoidance of patterns with incomparable elements. However, we do not believe pattern avoidance in poset permutations has been considered in the way we define below.A central theme in the study of pattern avoidance is demonstrating relationships between the number of permutations that avoid different patterns of the same length. In this paper, we define a notion of poset pattern avoidance and demonstrate one non-trivial inequality between avoidance of two length-three patterns. In particular, our work builds on a dichotomy between 1

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“…The permutations in this paper, with precisely two copies of each letter, are a special case of multiset permutations in which there may be an arbitrary numbers of copies of each letter. More detailed work with pattern avoidance involving multiset permutations can be found in [1], [3], [5], and [6].…”

Section: For Further Readingmentioning

confidence: 99%

Stacking Blocks and Counting Permutations

Pudwell

2010

Mathematics Magazine

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In this paper we will explore two seemingly unrelated counting questions, both of which are answered by the same formula. In the first section, we find the surface areas of certain solids formed from unit cubes. In second section, we enumerate permutations with a specified set of restrictions. Next, we give a bijection between the faces of the solids and the set of permutations. We conclude with suggestions for further reading. First, however, it is worth explaining how this paper came about.The author received an email from David Harris while he was helping his 12-year-old daughter complete a project for her math class. Together Harris and his daughter constructed triangular piles of cubes. After creating an increasing sequence of these piles, they computed the surface area of each pile, and hoped to find a formula for the surface area of their nth pile. This project and its solution are described in the next section. At the time of their correspondence, Harris and his daughter had deduced several facts about the construction but were unable to find a formula for the surface area in general. When they searched for the first few terms in their sequence, Google returned only one hit: a Maple data file on the author's website.The sequence that Harris and his daughter discovered online was originally generated in the context of pattern-avoiding words and permutations. Their web search produced a conjecture that gives a nice geometric interpretation of a permutation patterns question. This serendipitous discovery of the surprising and beautiful connection between a geometry problem and an enumeration problem illustrates how attractive new results may sometimes appear in such a surprising place as an elementary homework exercise.

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Avoiding patterns of length three in compositions and multiset permutations

Cited by 22 publications

References 3 publications

On the sub-permutations of pattern avoiding permutations

On the sub-permutations of pattern avoiding permutations

Pattern Avoidance in Poset Permutations

Stacking Blocks and Counting Permutations

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