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Discrete orthogonal polynomial ensembles and the Plancherel measure
, 2001
"... We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble i ..."
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Cited by 140 (8 self)
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We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a twodimensional directed growth model, and the Charlier ensemble is related to the lengths of weakly increasing subsequences in random words. The Krawtchouk ensemble occurs in connection with zigzag paths in random domino tilings of the Aztec diamond, and also in a certain simplified directed firstpassage percolation model. We use the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of Tracy and Widom. As a limit of the Meixner ensemble or the Charlier ensemble we obtain the Plancherel measure on partitions, and using this we prove a conjecture of Baik, Deift and Johansson that under the Plancherel measure, the distribution of the lengths of the first k rows in the partition, appropriately scaled, converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the Gaussian Unitary Ensemble. In this problem a certain discrete kernel, which we call the discrete Bessel kernel, plays an important role.
Asymptotics of Plancherel measures for symmetric groups
 J. Amer. Math. Soc
, 2000
"... 1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /G. For the symmetric group S(n), the set S(n) ∧ is the set o ..."
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Cited by 137 (33 self)
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1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /G. For the symmetric group S(n), the set S(n) ∧ is the set of partitions λ of the number
Generating Trees and the Catalan and Schröder Numbers
 DEPARTMENT OF MATHEMATICS, STOCKHOLMS UNIVERSITET, S106 91
, 1995
"... A permutation 2 Sn avoids the subpattern iff has no subsequence having all the same pairwise comparisons as , and we write 2 Sn ( ). We present a new bijective proof of the wellknown result that jS n (123)j = jS n (132)j = c n , the nth Catalan number. A generalization to forbidden pattern ..."
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Cited by 103 (3 self)
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A permutation 2 Sn avoids the subpattern iff has no subsequence having all the same pairwise comparisons as , and we write 2 Sn ( ). We present a new bijective proof of the wellknown result that jS n (123)j = jS n (132)j = c n , the nth Catalan number. A generalization to forbidden patterns of length 4 gives an asymptotic formula for the vexillary permutations. We settle a conjecture of Shapiro and Getu that jS n (3142; 2413)j = s n\Gamma1 , the Schröder number, and characterize the dequesortable permutations of Knuth, also counted by s n\Gamma1 .
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.
Excluded permutation matrices and the StanleyWilf conjecture
 J. Combin. Theory Ser. A
, 2004
"... This paper examines the extremal problem of how many 1entries an n × n 0–1 matrix can have that avoids a certain fixed submatrix P. For any permutation matrix P we prove a linear bound, settling a conjecture of Zoltán Füredi and Péter Hajnal [8]. Due to the work of Martin Klazar [12], this also set ..."
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Cited by 77 (3 self)
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This paper examines the extremal problem of how many 1entries an n × n 0–1 matrix can have that avoids a certain fixed submatrix P. For any permutation matrix P we prove a linear bound, settling a conjecture of Zoltán Füredi and Péter Hajnal [8]. Due to the work of Martin Klazar [12], this also settles the conjecture of Stanley and Wilf on the number of npermutations avoiding a fixed permutation and a related conjecture of Alon and Friedgut [1]. 1
On the distributions of the lengths of the longest monotone subsequences in random words
"... We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. (In the limit as k → ∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant rep ..."
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Cited by 50 (9 self)
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We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. (In the limit as k → ∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k×k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N → ∞ limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k × k hermitian matrices of trace zero. I.
The Asymptotics of Monotone Subsequences of Involutions
, 2001
"... We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the ..."
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Cited by 50 (5 self)
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We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the number of fixed points, (1) the TracyWidom distributions for the laxgest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of the authors in [7] which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the RiemannHilbert analysis for the orthogonal polynomials by Delft, Johansson and the first author in [3].
On the StanleyWilf conjecture for the number of permutations avoiding a given pattern
, 1999
"... . Consider, for a permutation oe 2 S k , the number F (n; oe) of permutations in Sn which avoid oe as a subpattern. The conjecture of Stanley and Wilf is that for every oe there is a constant c(oe) ! 1 such that for all n, F (n; oe) c(oe) n . All the recent work on this problem also mentions the ..."
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Cited by 46 (0 self)
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. Consider, for a permutation oe 2 S k , the number F (n; oe) of permutations in Sn which avoid oe as a subpattern. The conjecture of Stanley and Wilf is that for every oe there is a constant c(oe) ! 1 such that for all n, F (n; oe) c(oe) n . All the recent work on this problem also mentions the "stronger conjecture" that for every oe, the limit of F (n; oe) 1=n exists and is finite. In this short note we prove that the two versions of the conjecture are equivalent, with a simple argument involving subadditivity. We also discuss npermutations, containing all oe 2 S k as subpatterns. We prove that this can be achieved with n = k 2 , we conjecture that asymptotically n (k=e) 2 is the best achievable, and we present Noga Alon's conjecture that n (k=2) 2 is the threshold for random permutations. Mathematics Subject Classification: 05A05,05A16. 1. Introduction Consider, for a permutation oe 2 S k , the set A(n; oe) of permutations 2 S n which avoid oe as a subpattern, and it...