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86
Consecutive patterns in permutations
 Adv. in Appl. Math
, 2003
"... Abstract. In this paper we study the distribution of the number of occurrences of a permutation σ as a subword among all permutations in Sn. We solve the problem in several cases depending on the shape of σ by obtaining the corresponding bivariate exponential generating functions as solutions of cer ..."
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Cited by 38 (6 self)
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Abstract. In this paper we study the distribution of the number of occurrences of a permutation σ as a subword among all permutations in Sn. We solve the problem in several cases depending on the shape of σ by obtaining the corresponding bivariate exponential generating functions as solutions of certain linear differential equations with polynomial coefficients. Our method is based on the representation of permutations as increasing binary trees and on symbolic methods. 1.
Increasing and decreasing subsequences and their variants
 Proceedings of International Congress of Mathematical Society
, 2006
"... Abstract.We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2,..., ..."
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Cited by 37 (2 self)
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Abstract.We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2,...,n was obtained by VershikKerov and (almost) by LoganShepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions.
Multiavoidance of generalised patterns
 Discrete Math
, 2002
"... Abstract Recently, Babson and Steingr'imsson introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We investigate simultaneous avoidance of two or more 3patterns without internal dashes, that is, where ..."
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Cited by 34 (16 self)
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Abstract Recently, Babson and Steingr'imsson introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We investigate simultaneous avoidance of two or more 3patterns without internal dashes, that is, where the pattern corresponds to a contiguous subword in a permutation.
Asymptotic enumeration of permutations avoiding generalized patterns
 Advances in Applied Mathematics 36
, 2006
"... Abstract. Motivated by the recent proof of the StanleyWilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permuta ..."
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Cited by 30 (5 self)
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Abstract. Motivated by the recent proof of the StanleyWilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial results, showing that the number of permutations avoiding them grows faster than for classical patterns but more slowly than for consecutive patterns. 1.
Words Restricted by 3Letter Generalized Multipermutation Patterns
 ANNALS OF COMBINATORICS
, 2003
"... We find exact formulas and/or generating functions for the number of words avoiding 3letter generalized multipermutation patterns and find which of them are equally avoided. ..."
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Cited by 26 (16 self)
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We find exact formulas and/or generating functions for the number of words avoiding 3letter generalized multipermutation patterns and find which of them are equally avoided.
The insertion encoding of permutations
, 2005
"... We introduce the insertion encoding, an encoding of finite permutations. Classes of permutations whose insertion encodings form a regular language are characterized. Some necessary conditions are provided for a class of permutations to have insertion encodings that form a context free language. Appl ..."
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Cited by 24 (3 self)
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We introduce the insertion encoding, an encoding of finite permutations. Classes of permutations whose insertion encodings form a regular language are characterized. Some necessary conditions are provided for a class of permutations to have insertion encodings that form a context free language. Applications of the insertion encoding to the evaluation of generating functions for classes of permutations, construction of polynomial time algorithms for enumerating such classes, and the illustration of bijective equivalence between classes are demonstrated.
On some properties of permutation tableaux
, 2006
"... Abstract. We consider the relation between various permutation statistics and properties of permutation tableaux. We answer some of the questions of Steingrímsson and Williams [9], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We ..."
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Cited by 23 (1 self)
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Abstract. We consider the relation between various permutation statistics and properties of permutation tableaux. We answer some of the questions of Steingrímsson and Williams [9], in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We also consider and enumerate sets of permutation tableaux related to some pattern restrictions on permutations. 1.
BIJECTIONS FOR PERMUTATION TABLEAUX
"... In this paper we propose two bijections between permutation tableaux and permutations. These bijections show how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RLminima and pattern enumerations. We then use those bijections to define subcl ..."
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Cited by 21 (6 self)
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In this paper we propose two bijections between permutation tableaux and permutations. These bijections show how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RLminima and pattern enumerations. We then use those bijections to define subclasses of permutation tableaux that are in bijection with set partitions.
Enumerating permutations avoiding a pair of BabsonSteingrímsson patterns
 Ars Combinatorica
, 2005
"... Abstract. Babson and Steingrímsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of typ ..."
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Cited by 20 (4 self)
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Abstract. Babson and Steingrímsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1, 2) or (2, 1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. In the present paper we give a complete solution for the number of permutations avoiding a pair of patterns of type (1, 2) or (2, 1). We also conjecture the number of permutations avoiding the patterns in any set of three or more such patterns. 1.
Counting occurrences of some subword patterns
 Discr. Math. Theor. Comp. Sci
"... Abstract. We find generating functions for the number of strings (words) containing a specified number of occurrences of certain types of orderisomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified numbe ..."
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Cited by 19 (13 self)
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Abstract. We find generating functions for the number of strings (words) containing a specified number of occurrences of certain types of orderisomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3letter subword pattern. 1.