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In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers (with positive integer indices i and j) that is nonincreasing in both indices. This means that and for all i and j. Moreover, only finitely many of the may be nonzero. Plane partitions are a generalization of partitions of an integer. A plane partition may be represented visually by the placement of a stack of unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form The sum of a plane partition is

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  • Partición de plano (es)
  • Plane partition (en)
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  • En matemáticas y especialmente en combinatoria, una partición de plano es un conjunto bidimensional de enteros no negativos (con i,j enteros y positivos) que es no creciente para ambos índices. Esto significa que y para todo i y j. Además solo finitamente muchos de los son distintos de cero. Una partición de plano puede representarse gráficamente mediante una pila de cubos unitarios sobre el punto (i, j) en el plano, resultando un sólido tridimensional como el mostrado en la imagen. La suma de una partición de plano es: Por ejemplo, hay 6 particiones de plano con suma 3: (es)
  • In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers (with positive integer indices i and j) that is nonincreasing in both indices. This means that and for all i and j. Moreover, only finitely many of the may be nonzero. Plane partitions are a generalization of partitions of an integer. A plane partition may be represented visually by the placement of a stack of unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form The sum of a plane partition is (en)
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  • http://commons.wikimedia.org/wiki/Special:FilePath/BeispielPlanePartition.jpg
  • http://commons.wikimedia.org/wiki/Special:FilePath/CyclicallySymmetricPlanePartitions_M.jpg
  • http://commons.wikimedia.org/wiki/Special:FilePath/CyclicallySymmetricSelfCoplementary.jpg
  • http://commons.wikimedia.org/wiki/Special:FilePath/SelfComplementaryPlanePartition_M.jpg
  • http://commons.wikimedia.org/wiki/Special:FilePath/SymmetricPlanePartitions.jpg
  • http://commons.wikimedia.org/wiki/Special:FilePath/TotalSymmetricPlanePartitions.jpg
  • http://commons.wikimedia.org/wiki/Special:FilePath/TotallySymmetricSelfCoplementary.jpg
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  • Plane partition (en)
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  • PlanePartition (en)
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  • En matemáticas y especialmente en combinatoria, una partición de plano es un conjunto bidimensional de enteros no negativos (con i,j enteros y positivos) que es no creciente para ambos índices. Esto significa que y para todo i y j. Además solo finitamente muchos de los son distintos de cero. Una partición de plano puede representarse gráficamente mediante una pila de cubos unitarios sobre el punto (i, j) en el plano, resultando un sólido tridimensional como el mostrado en la imagen. La suma de una partición de plano es: La suma describe el número de cubos que forman la partición de plano. El número de particiones de plano de suma n se denota como PL(n). Por ejemplo, hay 6 particiones de plano con suma 3: Por lo que PL(3) = 6. (En este caso, las particiones de plano se obtienen usando indexado de matrices para las coordenadas y las entradas iguales a cero son eliminadas en pos de la legibilidad). Sea el número total de particiones de plano en las que r es el número de filas distintas de cero, s es el número de columnas distintas de cero y t es el mayor entero de la matriz. Las particiones de plano son a menudo descritas por las posiciones de los . Por ello una partición de plano se define como un subconjunto finito de puntos de latitud enteros positivos (i, j, k) en , tal que (r, s, t) están contenidos en y si (i, j, k) satisface , y , entonces (i, j, k) también están contenidos en . (es)
  • In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers (with positive integer indices i and j) that is nonincreasing in both indices. This means that and for all i and j. Moreover, only finitely many of the may be nonzero. Plane partitions are a generalization of partitions of an integer. A plane partition may be represented visually by the placement of a stack of unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subset of positive integer lattice points (i, j, k) in , such that if (r, s, t) lies in and if satisfies , , and , then (i, j, k) also lies in . The sum of a plane partition is The sum describes the number of cubes of which the plane partition consists. Much interest in plane partitions concerns the enumeration of plane partitions in various classes. The number of plane partitions with sum n is denoted by PL(n). For example, there are six plane partitions with sum 3 so PL(3) = 6. Plane partitions may be classified by how symmetric they are. Many symmetric classes of plane partitions are enumerated by simple product formulas. (en)
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