Generating trees and the Catalan and Schröder numbers

West, Julian

doi:10.1016/0012-365x(94)00067-1

Cited by 205 publications

(82 citation statements)

References 12 publications

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“…4,5] and the special issue [9]. Call two permutations Let us mention one useful technique for showing the equivalence of permutations in S k , the method of generating trees introduced by Chung, Graham, Hoggatt and Kleiman [38] and further developed by West [111], [112], [113] and others. Given u ∈ S k , the generating tree T u is the tree with vertex set n≥1 S n (u), and with y a descendent of w if w is a subsequence of y (an actual subsequence, not the pattern of a subsequence).…”

Section: Pattern Avoidancementioning

confidence: 99%

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Increasing and decreasing subsequences and their variants

Stanley¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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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 Vershik-Kerov and (almost) by Logan-Shepp. 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. Mathematics Subject Classification (2000). Primary 05A05, 05A06; Secondary 60C05.

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Section: Pattern Avoidancementioning

confidence: 99%

“…They turn out to be the same as 2143-avoiding permutations [73, (1.27)(iii), (7.24)(iii)], first enumerated by West [111,Cor. 3.17], [112,Cor. 3.11].…”

Section: Pattern Avoidancementioning

confidence: 99%

Increasing and decreasing subsequences and their variants

Stanley¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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“…Therefore a single object produces a set of new objects according to some parameter. Typically, if such a construction is regular enough, one can encode it using a succession rule [32], which is a purely formal system, generally expressed as follows:…”

Section: Introductionmentioning

confidence: 99%

A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

Barcucci

Bernini

Ferrari³

et al. 2005

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In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference 135 (2005), 77-92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math. 217 (2000), 367-409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.

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“…= {321, 3412, 4123}, χ T (i, k) = 3 if i = 1 2 otherwise (iii) even-index Fibonacci numbers[4].-T = {321, 3412}, χ T (i, k) = k + 1 if i = 1 2 otherwise -T = {321, 4123}, χ T (i, k) = 3 if i = 1 i otherwise (iv) Catalan numbers[17,25].-T = {312}, χ T (i, k) = i + 1 -T = {321}, χ T (i, k) = k + 1 if i = 1 i otherwise (v) Schröder numbers[11].-T = {4321, 4312}, χ T (i, k) = k + 1 if i = 1 or i = 2 i otherwise -T = {4231, 4132}, χ T (i, k) = k + 1 if i = 1 or i = k i + 1 otherwise -T = {4123, 4213}, χ T (i, k) = k + 1 if i = k − 1 or i = k i + 2 otherwise(vi) central binomial coefficients 2n−2 n−1 [11]. -T = {4321, 4231, 4312, 4132}, χ T (i, T = {4231, 4132, 4213, 4123}, χ T (i, k) = 3 if i = 1 i + 1 otherwise.…”

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confidence: 99%

Combinatorial Gray codes for classes of pattern avoiding permutations

Dukes¹,

Flanagan²,

Mansour³

et al. 2008

Theoretical Computer Science

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The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, large Schröder, Pell, even-index Fibonacci numbers and the central binomial coefficients. We thus provide Gray codes for the set of all permutations of {1, . . . , n} avoiding the pattern τ for all τ ∈ S 3 and the Gray codes we obtain have distances 4 or 5.

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Generating trees and the Catalan and Schröder numbers

Cited by 205 publications

References 12 publications

Increasing and decreasing subsequences and their variants

Increasing and decreasing subsequences and their variants

A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

Combinatorial Gray codes for classes of pattern avoiding permutations

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