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- En las matemáticas relacionadas con la combinatoria, un polinomio de torre, es una expresión numérica que genera el número de formas de colocar una serie de torres que no se atacan entre sí en un tablero similar a uno de ajedrez, aunque de dimensión variable. No existe la posibilidad de que dos torres se encuentren en la misma fila o columna. El tablero es cualquier subconjunto de las casillas de un tablero rectangular con m filas y n columnas. Coincide con el tablero de ajedrez común si cualquier casilla puede ser ocupada por una pieza y m=n=8, y m=n. El coeficiente de xk en el polinomio torre RB(x), es el número de maneras en que las k torres (ninguna de las cuales ataca a otra), pueden ser dispuestas en los cuadrados del tablero B. Las torres están dispuestas de tal manera que no hay dos torres en la misma fila o columna. En este sentido, una forma de colocarlas es situar las torres en una tabla estática e inmóvil. La configuración no será diferente mientras se mantengan las casillas fijas. El polinomio también permanece igual, aunque se cambien filas por columnas. (es)
- In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged in such a way that there is no pair of rooks in the same row or column. In this sense, an arrangement is the positioning of rooks on a static, immovable board; the arrangement will not be different if the board is rotated or reflected while keeping the squares stationary. The polynomial also remains the same if rows are interchanged or columns are interchanged. The term "rook polynomial" was coined by John Riordan.Despite the name's derivation from chess, the impetus for studying rook polynomials is their connection with counting permutations (or partial permutations) with restricted positions. A board B that is a subset of the n × n chessboard corresponds to permutations of n objects, which we may take to be the numbers 1, 2, ..., n, such that the number aj in the j-th position in the permutation must be the column number of an allowed square in row j of B. Famous examples include the number of ways to place n non-attacking rooks on:
* an entire n × n chessboard, which is an elementary combinatorial problem;
* the same board with its diagonal squares forbidden; this is the derangement or "hat-check" problem (this is a particular case of the problème des rencontres);
* the same board without the squares on its diagonal and immediately above its diagonal (and without the bottom left square), which is essential in the solution of the problème des ménages. Interest in rook placements arises in pure and applied combinatorics, group theory, number theory, and statistical physics. The particular value of rook polynomials comes from the utility of the generating function approach, and also from the fact that the zeroes of the rook polynomial of a board provide valuable information about its coefficients, i.e., the number of non-attacking placements of k rooks. (en)
- 루크 다항식(Rook Polynomials)은 이산 수학에 소개되어있는 내용으로, 체스의 말 중 하나인 루크를 사용해 만든 다항식 문제이다. 체스의 말 중 하나인 룩은 마치 장기의 차(車)와 같이 자신이 놓인 행 또는 열에 있는 다른 어떤 말도 직선으로 가서 잡는다. 루크 다항식은 이 룩들이 서로 잡지 못하도록 체스판 위의 제한된 곳에 놓는 방법의 수를 구하는 문제이다. 체스를 이용한 다른 문제로는 퀸 문제, 나이트 문제 등등이 있다. 현재 n곱하기n꼴에서의 룩의 최대 개수에 대한 일반항을 연구하는 수학자들도 존재한다. (ko)
- 组合数学的核心是解决计数问题，其中很重要的即为n个元素的排列方案的计数。一个常见的将排列问题抽象的方法就是将其抽象为棋盘多项式。首先看一个的棋盘，n个元素的排列可以看成在这个棋盘上落下n个棋子，其中每一个横行、每一个竖列只允许有一个棋子。而其中棋盘的格子是可以任意的的棋盘的子集，这对应了存在一定限制的排列方案。 每一个棋盘对应着一个母函数代表该棋盘中描述无法攻击的棋子排列数。这个母函数即为棋盘多项式。 (zh)
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- 루크 다항식(Rook Polynomials)은 이산 수학에 소개되어있는 내용으로, 체스의 말 중 하나인 루크를 사용해 만든 다항식 문제이다. 체스의 말 중 하나인 룩은 마치 장기의 차(車)와 같이 자신이 놓인 행 또는 열에 있는 다른 어떤 말도 직선으로 가서 잡는다. 루크 다항식은 이 룩들이 서로 잡지 못하도록 체스판 위의 제한된 곳에 놓는 방법의 수를 구하는 문제이다. 체스를 이용한 다른 문제로는 퀸 문제, 나이트 문제 등등이 있다. 현재 n곱하기n꼴에서의 룩의 최대 개수에 대한 일반항을 연구하는 수학자들도 존재한다. (ko)
- 组合数学的核心是解决计数问题，其中很重要的即为n个元素的排列方案的计数。一个常见的将排列问题抽象的方法就是将其抽象为棋盘多项式。首先看一个的棋盘，n个元素的排列可以看成在这个棋盘上落下n个棋子，其中每一个横行、每一个竖列只允许有一个棋子。而其中棋盘的格子是可以任意的的棋盘的子集，这对应了存在一定限制的排列方案。 每一个棋盘对应着一个母函数代表该棋盘中描述无法攻击的棋子排列数。这个母函数即为棋盘多项式。 (zh)
- En las matemáticas relacionadas con la combinatoria, un polinomio de torre, es una expresión numérica que genera el número de formas de colocar una serie de torres que no se atacan entre sí en un tablero similar a uno de ajedrez, aunque de dimensión variable. No existe la posibilidad de que dos torres se encuentren en la misma fila o columna. El tablero es cualquier subconjunto de las casillas de un tablero rectangular con m filas y n columnas. Coincide con el tablero de ajedrez común si cualquier casilla puede ser ocupada por una pieza y m=n=8, y m=n. (es)
- In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged in such a way that there is no pair of rooks in the same row or column. In this sense, an arrangem (en)
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