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J.Propp, The shape of a typical boxed plane partition
 J. of Math
, 1998
"... Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the ..."
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Cited by 51 (5 self)
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Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux. 1.
Generating Random Elements of Finite Distributive Lattices
 Electronic Journal of Combinatorics
, 1997
"... This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using "coupling from the past" to remove initialization bias from Monte ..."
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Cited by 18 (1 self)
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This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using "coupling from the past" to remove initialization bias from Monte Carlo randomization.
Binomial Identities  Combinatorial and Algorithmic Aspects
, 1994
"... The problem of proving a particular binomial identity is taken as an opportunity to discuss various aspects of this field and to discuss various proof techniques in an examplary way. In particular, the unifying role of the hypergeometric nature of binomial identities is underlined. This aspect is al ..."
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Cited by 15 (3 self)
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The problem of proving a particular binomial identity is taken as an opportunity to discuss various aspects of this field and to discuss various proof techniques in an examplary way. In particular, the unifying role of the hypergeometric nature of binomial identities is underlined. This aspect is also basic for combinatorial models and techniques, developed during the last decade, and for the recent algorithmic proof procedures. Much of mathematics comes from looking at very simple examples from a more general perspective. Hypergeometric functions are a good example of this. R. Askey 1 Introduction In this article I want to highlight some aspects of "binomial identites" or "combinatorial sums" in an exemplary way. Writing such an article was motivated by a question that I was asked in spring 1992, and by my subsequent investigations on it: Can you show that the binomial identity n X k=0 ` n k ' 2 ` n + k k ' 2 = n X k=0 ` n k '` n + k k ' k X j=0 ` k j ' 3 (1)...
Enumeration of matchings: problems and progress
 in New Perspectives in Algebraic Combinatorics
, 1999
"... Abstract. This document is built around a list of thirtytwo problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary ..."
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Cited by 5 (0 self)
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Abstract. This document is built around a list of thirtytwo problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and online literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems. 1.
More Statistics on Permutation Pairs
 Electronic Journal of Combinatorics
, 1994
"... Two inversion formulas for enumerating words in the free monoid by `adjacencies are applied in counting pairs of permutations by various statistics. The generating functions obtained involve refinements of bibasic Bessel functions. We further extend the results to finite sequences of permutations. ..."
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Cited by 4 (0 self)
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Two inversion formulas for enumerating words in the free monoid by `adjacencies are applied in counting pairs of permutations by various statistics. The generating functions obtained involve refinements of bibasic Bessel functions. We further extend the results to finite sequences of permutations. This work is partially supported by EC grant CHRXCT930400 and PRC MathsInfo y Financial support provided by LaBRI, Universit'e Bordeaux I the electronic journal of combinatorics 1 (1994), #R11 1 1 Introduction The study of statistics on permutation pairs was initiated by Carlitz, Scoville, and Vaughan [4]. Stanley [18] qextended their work to finite sequences of permutations. In [6], we exploited the recursive technique of Carlitz et. al. to obtain some additional refinements. We also discussed numerous related distributions. Our purpose here is to further extend the study of statistics on finite permutation sequences. Our method is based on the theory of inversion presented in ...
Enumeration of Matchings
, 1998
"... : This document is built around a list of twenty problems in enumeration of matchings that were gathered together in 1996 and presented in a lecture at MSRI that fall. Since then, roughly half of the problems have been solved by participants in the MSRI Workshop on Combinatorics, their students, and ..."
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Cited by 2 (0 self)
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: This document is built around a list of twenty problems in enumeration of matchings that were gathered together in 1996 and presented in a lecture at MSRI that fall. Since then, roughly half of the problems have been solved by participants in the MSRI Workshop on Combinatorics, their students, and others. The article begins with a capsule history of the topic of enumeration of matchings. The twenty problems themselves, with commentary, comprise the bulk of the article. The final section gives an account of the progress that has been made on these problems as of this writing, and includes pointers to both the printed and online literature. 1 Introduction How many perfect matchings does a given graph G have? That is, in how many ways can one choose a subset of the edges of G so that each vertex of G belongs to one and only one chosen edge? (See Figure 1(a) for an example of a matching of a graph. The book by Lov'asz and Plummer [LP] gives general background on matchings of graphs.) ...
A Geometric Proof of a Formula for the Number of Young Tableaux of a Given Shape
, 1996
"... This paper contains a short proof of a formula by Frame, Robinson, and Thrall [1] which counts the number of Young tableaux of a given shape. The proof is based on a simple but novel geometric way of expressing the area of a Ferrers diagram. research supported in part by National Science Foundatio ..."
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This paper contains a short proof of a formula by Frame, Robinson, and Thrall [1] which counts the number of Young tableaux of a given shape. The proof is based on a simple but novel geometric way of expressing the area of a Ferrers diagram. research supported in part by National Science Foundation operating grant CCR9304722. ii 1 Introduction Let = f 1 2 \Delta \Delta \Delta m g be a partition of n. The Ferrers diagram of is an array of cells indexed by pairs (i; j) with 1 i m, 1 j i . A Young tableau of shape (sometimes called a standard tableau) is an arrangement of the integers 1; 2; : : : ; n in the cells of the Ferrers diagram of such that all rows and columns form increasing sequences. The total number of Young tableaux of shape will be denoted f(). For each cell (i; j) define the hook H i;j to be the collection of cells (a; b) such that a = i and b j or a i and b = j. Define the hook length h i;j to be the number of cells in H i;j . (See Figure 1.) 1 2...
Additive Decompositions, Random Allocations, and Threshold Phenomena
, 2003
"... An additive decomposition of a set I of nonnegative integers is an expression of I as the arithmetic sum of two other such sets. If the smaller of these has p elements, we have a pdecomposition. If I is obtained ..."
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An additive decomposition of a set I of nonnegative integers is an expression of I as the arithmetic sum of two other such sets. If the smaller of these has p elements, we have a pdecomposition. If I is obtained
167 168 ALGORITHMS FOR THIRDORDER RECURSION SEQUENCES [April
"... Given a thirdorder recursion relation ..."