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27
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.
Enumeration of perfect matchings in graphs with reflective symmetry
 J. Combin. Theory Ser. A
, 1997
"... Abstract. A plane graph is called symmetric if it is invariant under the reflection across some straight line. We prove a result that expresses the number of perfect matchings of a large class of symmetric graphs in terms of the product of the number of matchings of two subgraphs. When the graph is ..."
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Cited by 49 (14 self)
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Abstract. A plane graph is called symmetric if it is invariant under the reflection across some straight line. We prove a result that expresses the number of perfect matchings of a large class of symmetric graphs in terms of the product of the number of matchings of two subgraphs. When the graph is also centrally symmetric, the two subgraphs are isomorphic and we obtain a counterpart of Jockusch’s squarishness theorem. As applications of our result, we enumerate the perfect matchings of several families of graphs and we obtain new solutions for the enumeration of two of the ten symmetry classes of plane partitions (namely, transposed complementary and cyclically symmetric, transposed complementary) contained in a given box. Finally, we consider symmetry classes of perfect matchings of the Aztec diamond graph and we solve the previously open problem of enumerating the matchings that are invariant under a rotation by 90 degrees. The starting point of this paper is a result [18, Theorem 1] concerning domino tilings of the Aztec diamond compatible with certain barriers. This result has also been generalized and proved bijectively by Propp [17]. We present (see Lemma 1.1) a further generalization,
Advanced determinant calculus: a complement
 Linear Algebra Appl
"... Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particu ..."
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Cited by 49 (6 self)
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Abstract. This is a complement to my previous article “Advanced Determinant Calculus ” (Séminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12 (2003), 441– 456). Moreover, I add a list of determinant evaluations which I consider as interesting, which have been found since the appearance of the previous article, or which I failed to mention there, including several conjectures and open problems. 1.
Advanced Determinant Calculus
, 1999
"... The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have ..."
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Cited by 37 (0 self)
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The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.
The number of rhombus tilings of a symmetric hexagon which contain a fixed rhombus on the symmetry axis
"... Abstract. We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M. 1. ..."
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Cited by 23 (7 self)
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Abstract. We compute the number of rhombus tilings of a hexagon with sides N, M, N, N, M, N, which contain a fixed rhombus on the symmetry axis that cuts through the sides of length M. 1.
Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole
"... . We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (\Gamma1)enume ..."
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Cited by 23 (9 self)
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. We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (\Gamma1)enumeration of these lozenge tilings. In the case that a = b = c, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For m = 0, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants det 0i;jn\Gamma1 \Gamma !ffi ij + \Gamma m+i+j j \Delta\Delta , w...
Plane partitions I: A generalization of MacMahon’s formula
 Memoirs Amer. Math. Soc
"... Abstract. The number of plane partitions contained in a given box was shown by MacMahon [8] to be given by a simple product formula. By a simple bijection, this formula also enumerates lozenge tilings of hexagons of sidelengths a, b, c, a, b, c (in cyclic order) and angles of 120 degrees. We presen ..."
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Cited by 14 (7 self)
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Abstract. The number of plane partitions contained in a given box was shown by MacMahon [8] to be given by a simple product formula. By a simple bijection, this formula also enumerates lozenge tilings of hexagons of sidelengths a, b, c, a, b, c (in cyclic order) and angles of 120 degrees. We present a generalization in the case b = c by giving simple product formulas enumerating lozenge tilings of the regions obtained from a hexagon of sidelengths a, b + k, b, a + k, b, b + k (where k is an arbitrary nonnegative integer) and angles of 120 degrees by removing certain triangular regions along its symmetry axis. 1.
The Number of Centered Lozenge Tilings of a Symmetric Hexagon
"... Propp conjectured [15] that the number of lozenge tilings of a semiregular hexagon of sides 2n \Gamma 1, 2n \Gamma 1 and 2n which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregu ..."
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Cited by 14 (7 self)
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Propp conjectured [15] that the number of lozenge tilings of a semiregular hexagon of sides 2n \Gamma 1, 2n \Gamma 1 and 2n which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we consider the more general situation of a semiregular hexagon of sides a, a and b. We prove explicit formulas for the number of lozenge tilings of these hexagons containing the central unit rhombus, and obtain Propp's conjecture as a corollary of our results.
Enumeration of lozenge tilings of hexagons with cut off corners
 J. Comb. Th. Ser. A
"... Abstract. Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with “maximal staircases ” removed from some of its vertices. The case of one ..."
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Cited by 13 (7 self)
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Abstract. Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with “maximal staircases ” removed from some of its vertices. The case of one vertex corresponds to Proctor’s problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose the problem of finding a formula for the number of tilings in this case.
Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the centre
"... Abstract. We compute the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the central rhombus and the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the ‘almost central ’ rhombus above the centre. ..."
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Cited by 9 (0 self)
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Abstract. We compute the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the central rhombus and the number of rhombus tilings of a hexagon with side lengths a,b,c,a,b,c which contain the ‘almost central ’ rhombus above the centre.