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Restricted 132avoiding permutations
 Adv. in Appl. Math
"... Abstract. We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second ..."
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Cited by 44 (23 self)
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Abstract. We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.
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.
Restricted 132alternating permutations and Chebyshev polynomials
 Annals of Combinatorics
"... A permutation is said to be alternating if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on n letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary p ..."
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Cited by 15 (1 self)
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A permutation is said to be alternating if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on n letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary pattern on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.
RESTRICTED 132 PERMUTATIONS AND GENERALIZED PATTERNS
, 2001
"... Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on n letters avoiding 132 (or containing 1 ..."
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Cited by 14 (6 self)
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Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on n letters avoiding 132 (or containing 132 exactly once) and an arbitrary generalized pattern τ on k letters, or containing τ exactly once. In several cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind, and generating function of Motzkin numbers. 1
Wilf classes of pairs of permutations of length 4
 Electronic J. Combinatorics
"... Sn(π1,π2,...,πr) denotes the set of permutations of length n that have no subsequence with the same order relations as any of the πi. In this paper we show that Sn(1342, 2143)  = Sn(3142, 2341)  and Sn(1342, 3124)  = Sn(1243, 2134). These two facts complete the classification of Wilfequiv ..."
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Cited by 14 (0 self)
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Sn(π1,π2,...,πr) denotes the set of permutations of length n that have no subsequence with the same order relations as any of the πi. In this paper we show that Sn(1342, 2143)  = Sn(3142, 2341)  and Sn(1342, 3124)  = Sn(1243, 2134). These two facts complete the classification of Wilfequivalence classes for pairs of permutations of length four. In both instances we exhibit bijections between the sets using the idea of a “block”, and in the former we find a generating function for Sn(1342, 2143). 1
Continued fractions, statistics, and generalized patterns
, 2001
"... Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following [BCS], let ekπ (respectively; fkπ) be the number of the occurrences of the generalized pattern 1 ..."
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Cited by 11 (7 self)
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Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Following [BCS], let ekπ (respectively; fkπ) be the number of the occurrences of the generalized pattern 123...k (respectively; 213...k) in π. In the present note, we study the distribution of the statistics ekπ and fkπ in a permutation avoiding the classical pattern 132. Also we present an applications, which relates the Narayana numbers, Catalan numbers, and increasing subsequences, to permutations avoiding the classical pattern 132 according to a given statistics on ekπ, or on fkπ.
Statistics on patternavoiding permutations
, 2004
"... This thesis concerns the enumeration of patternavoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics ‘number of fixed points’ and ‘number of excedances’ is the same in 321avoiding as in 132avoiding permutations. This ge ..."
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Cited by 10 (1 self)
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This thesis concerns the enumeration of patternavoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics ‘number of fixed points’ and ‘number of excedances’ is the same in 321avoiding as in 132avoiding permutations. This generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof. The key ideas are to introduce a new class of statistics on Dyck paths, based on what we call a tunnel, and to use a new technique involving diagonals of nonrational generating functions. Next we present a new statisticpreserving family of bijections from the set of Dyck paths to itself. They map statistics that appear in the study of patternavoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. In particular, this gives a simple bijective proof of the equidistribution of fixed points in the above two sets of restricted permutations. Then we introduce a bijection between 321 and 132avoiding permutations that preserves
WHEN IS A SCHUBERT VARIETY GORENSTEIN?
, 2004
"... The main goal of this paper is to give an explicit combinatorial characterization of which Schubert varieties in the complete flag variety are Gorenstein. Let Flags(C n) denote the variety of complete flags F • : 〈0 〉 ⊆ F1 ⊆... ⊆ Fn = C n. Fix a basis e1, e2,..., en of C n and let E • be the antic ..."
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Cited by 10 (1 self)
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The main goal of this paper is to give an explicit combinatorial characterization of which Schubert varieties in the complete flag variety are Gorenstein. Let Flags(C n) denote the variety of complete flags F • : 〈0 〉 ⊆ F1 ⊆... ⊆ Fn = C n. Fix a basis e1, e2,..., en of C n and let E • be the anticanonical reference flag E•, that is, the flag
Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials
 DISC. MATH
, 2005
"... We say that a permutation π is a Motzkin permutation if it avoids 132 and there do not exist a < b such that πa < πb < πb+1. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of ..."
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Cited by 10 (5 self)
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We say that a permutation π is a Motzkin permutation if it avoids 132 and there do not exist a < b such that πa < πb < πb+1. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of rises and descents. We also enumerate Motzkin permutations with additional restrictions, and study the distribution of occurrences of fairly general patterns in this class of permutations.