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20
Longest increasing subsequences: from patience sorting to the BaikDeiftJohansson theorem
 Bull. Amer. Math. Soc. (N.S
, 1999
"... Abstract. We describe a simple oneperson card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of BaikDeiftJoha ..."
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Cited by 137 (2 self)
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Abstract. We describe a simple oneperson card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of BaikDeiftJohansson which yields limiting probability laws via hard analysis of Toeplitz determinants. 1.
Exact Enumeration Of 1342Avoiding Permutations A Close Link With Labeled Trees And Planar Maps
, 1997
"... Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of inde ..."
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Cited by 84 (7 self)
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Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of indecomposable 1342avoiding permutations of length n equals that of labeled plane trees of a certain type on n vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, H(x) turns out to be algebraic, proving the first nonmonotonic, longerthanthree instance of a conjecture of Zeilberger and Noonan. We also prove that n p Sn (1342) converges to 8, so in particular, limn!1 (Sn (1342)=Sn (1234)) = 0.
On the nonholonomic character of logarithms, powers, and the nth prime function
, 2005
"... We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction ..."
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Cited by 16 (6 self)
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We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularitybased methods and several naturally occurring sequences are proved to be nonholonomic.
Noncrossing tworowed arrays and summations for Schur functions
 Proc. of the 5th Conference on Formal Power Series and Algebraic Combinatorics
, 1993
"... In the first part of this paper (sections 1,2) we give combinatorial proofs for determinantal formulas for sums of Schur functions "in a strip" that were originally obtained by Gessel, respectively Goulden, using algebraic methods. The combinatorial analysis involves certain families of tworowed ar ..."
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Cited by 7 (5 self)
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In the first part of this paper (sections 1,2) we give combinatorial proofs for determinantal formulas for sums of Schur functions "in a strip" that were originally obtained by Gessel, respectively Goulden, using algebraic methods. The combinatorial analysis involves certain families of tworowed arrays, asymmetric variations of Sagan and Stanley's skew Knuthcorrespondence, and variations of one of Burge's correspondences. In the third section we specialize the parameters in these determinants to compute norm generating functions for tableaux in a strip. In case we can get rid of the determinant we obtain multifold summations that are basic hypergeometric series for Ar and Cr respectively. In some cases these sums can be evaluated. Thus in particular, an alternative proof for refinements of the BenderKnuth and MacMahon (ex)Conjectures, which were first obtained in another paper by the author, is provided. Although there are some parallels with the original proof, perhaps this proof is easier accessible. Finally, in section 4, we record further applications of our methods to the enumeration of paths with respect to weighted turns. 1. Generating functions for noncrossing tworowed arrays. We consider tworowed arrays P = (p j q) of the form p \Gammaa p \Gammaa+1 : : : p \Gamma1 p 1 : : : p k q 1 : : : q k q \Gamma1 : : : q \Gammab+1 q \Gammab ; (1.1) where a; k; b are some nonnegative integers and where the entries p i ; q i are positive integers such that both rows of the array are weakly increasing. (To be precise, if k = 0, i.e. the "middle part" of the array is empty, for a c minfa; bg we also allow the entries p \Gamma1 ; : : : ; p \Gammac and q \Gamma1 ; : : : ; q \Gammac to be "empty".) We say that P is of the type (a; b) and of the shape (a; k; b). If bo...
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 ..."
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Cited by 6 (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
A survey on certain pattern problems
"... Abstract. The paper contains all the definitions and notations needed to understand the results concerning the field dealing with occurrences of patterns in permutations and words. Also, this paper includes a historical overview on the results obtained in this subject. The authors tried to collect a ..."
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Cited by 4 (3 self)
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Abstract. The paper contains all the definitions and notations needed to understand the results concerning the field dealing with occurrences of patterns in permutations and words. Also, this paper includes a historical overview on the results obtained in this subject. The authors tried to collect all the currently existing references to the papers directly related to the subject. Moreover, a number of basic approaches to study the pattern problems are discussed.
Lattice walks in Z^d and permutations with no long ascending subsequences
, 1997
"... We identify a set of d! signed points, called Toeplitz points,inZ d , with the following property: for every n>0, the excess of the number of lattice walks of n steps, from the origin to all positive Toeplitz points, over the number to all negative Toeplitz points, is equal to # n n/2 # times ..."
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Cited by 3 (0 self)
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We identify a set of d! signed points, called Toeplitz points,inZ d , with the following property: for every n>0, the excess of the number of lattice walks of n steps, from the origin to all positive Toeplitz points, over the number to all negative Toeplitz points, is equal to # n n/2 # times the number of permutations of {1, 2,...,n} that contain no ascending subsequence of length >d. We prove this first by generating functions, using a determinantal theorem of Gessel. We give a second proof by direct construction of an appropriate involution. The latter provides a purely combinatorial proof of Gessel's theorem by interpreting it in terms of lattice walks. Finally we give a proof that uses the Schensted algorithm. Submitted: September 27, 1996; Accepted: November 17, 1997 1 Introduction The subject of walks on the lattice in Euclidean space is one of the most important areas of combinatorics. Another subject that has been investigated by many researchers in recent # Support...