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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.
Applications of graphical condensation for enumerating matchings and tilings, arXiv: math.CO/0304090
"... A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then repar ..."
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Cited by 29 (0 self)
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A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then repartitioning the united matching (actually a multigraph) into matchings of two other subgraphs, in one of two possible ways. This technique can be used to enumerate perfect matchings of a wide variety of bipartite planar graphs. Applications include domino tilings of Aztec diamonds and rectangles, diabolo tilings of fortresses, plane partitions, and transpose complement plane partitions. 1
Perfect matchings and the octahedron recurrence
 math.CO/0402452, 2004. André Henriques, Mathematisches Institut, Westfälische WilhelmsUniversität, Einsteinstr. 62, 48149
"... We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conje ..."
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Cited by 27 (1 self)
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We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky. 1
Characterizations of FlipAccessibility for Domino Tilings of the Whole Plane
"... Abstract. It is known that any two domino tilings of a polygon are flipaccessible, i.e., linked by a finite sequence of local transformations, called flips. This paper considers flipaccessibility for domino tilings of the whole plane, asking whether two of them are linked by a possibly infinite se ..."
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Abstract. It is known that any two domino tilings of a polygon are flipaccessible, i.e., linked by a finite sequence of local transformations, called flips. This paper considers flipaccessibility for domino tilings of the whole plane, asking whether two of them are linked by a possibly infinite sequence of flips. The answer turning out to depend on tilings, we provide three equivalent characterizations of flipaccessibility. Résumé. Étant donnés deux pavages par dominos d’un mme polygone, on sait qu’on peut toujours passer de l’un l’autre en effectuant un nombre fini de transformations locales, appelées flips; ces pavages sont dits flipaccessibles. Dans ce papier, nous étendons cette notion de flipaccessibilité aux pavages par dominos du plan entier, en s’autorisant cette fois effectuer un nombre infini de flips. Dans ce cas, la flipaccessibilité dépend des pavages considérés et nous en donnons trois caractérisations équivalentes.