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Generating polynomial of the number of ways to place non-attacking rooks on a chessboard
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
Rook_polynomial
Counting technique in combinatorics
to calculate the highest coefficient of a rook polynomial in terms of the coefficients of the rook polynomial of the complementary board. Without loss
Inclusion–exclusion_principle
Graph polynomial generating numbers of matchings
The first type of matching polynomial is a direct generalization of the rook polynomial. The second type of matching polynomial has remarkable connections
Matching_polynomial
Sequence of differential equation solutions
_{0}^{\infty }f(x)g(x)e^{-x}\,dx.} The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables
Laguerre_polynomials
polynomials Rogers polynomials Rogers–Szegő polynomials Rook polynomial Schur polynomials Shapiro polynomials Sheffer sequence Spread polynomials Tricomi–Carlitz
List_of_polynomial_topics
Sequence valued in polynomials
polynomials Lucas polynomials Spread polynomials Touchard polynomials Rook polynomials Polynomial sequences of binomial type Orthogonal polynomials Secondary
Polynomial_sequence
Mathematical problem set on a chessboard
array Mathematical game Mathematical puzzle No-three-in-line problem Rook polynomial The number of combinations of 8 squares from 64 is the binomial coefficient
Eight_queens_puzzle
Graph of chess rook moves
theory, a rook's graph is an undirected graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents
Rook's_graph
Polynomial of the elements of a matrix
(These numbers arise in combinatorics as leading coefficients of rook polynomials.) The Bregman–Minc inequality, conjectured by H. Minc in 1963 and proved
Permanent_(mathematics)
Matrix with exactly one 1 per row and column
Alternating sign matrix Exchange matrix Generalized permutation matrix Rook polynomial Permanent Cayley graph – graph constructed from a group whose adjacency
Permutation_matrix
Formal power series
enumeration problems in combinatorics and encoding their solutions. Rook polynomials are an example of an application in combinatorics. Evaluate infinite
Generating_function
Patience or card solitaire
mathematics, which can be understood using a combinatorial tool called a rook polynomial. The probability of winning the game has been determined exactly, and
Frustration_(card_game)
Game in combinatorics
1006/eujc.1994.1057, MR 1302079. Spivey, Michael Z. (2009), "Staircase rook polynomials and Cayley's game of Mousetrap" (PDF), European Journal of Combinatorics
Cayley's_mousetrap
Number of ways to pair up n objects
involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with n cells, and the sum of the
Telephone number (mathematics)
Telephone_number_(mathematics)
Undirected graph acted on by a vertex-transitive cyclic group of symmetries
then the m × n rook's graph (a graph that has a vertex for each square of an m × n chessboard and an edge for each two squares that a rook can move between
Circulant_graph
Set of matrices
matrices, these polynomial problems with restricted coefficients can be framed as Bohemian matrix problems. However, the characteristic polynomial of a Bohemian
Bohemian_matrices
Graph with equal-size maximal independent sets
bipartite graphs, and the rook's graphs whose vertices represent squares of a chessboard and edges represent moves of a chess rook. Known characterizations
Well-covered_graph
American mathematician
1993 from the University of Georgia, with the dissertation Compositions, Rook Placements, and Permutations of Vectors supervised by Earl Rodney Canfield
James_Haglund
Operation in graph theory
representation has the same number of irreducible factors. Imrich (1998) gives a polynomial time algorithm for recognizing tensor product graphs and finding a factorization
Tensor_product_of_graphs
Operation in graph theory
vertices and edges of an n-prism is the Cartesian product graph K2□Cn. The rook's graph is the Cartesian product of two complete graphs. If a connected graph
Cartesian_product_of_graphs
Curve used in computer graphics and related fields
mathematical basis for Bézier curves—the Bernstein polynomials—was established in 1912, but the polynomials were not applied to graphics until some 50 years
Bézier_curve
Graph which is isomorphic to its complement
graph. Every Paley graph is self-complementary. For example, the 3 × 3 rook's graph (the Paley graph of order nine) is self-complementary, by a symmetry
Self-complementary_graph
Problem in graph theory
the rook's graphs, for which it corresponds to the problem of completing a partially filled-in Latin square. The problem may be solved in polynomial time
Precoloring_extension
Undirected graph named after S. S. Shrikhande
Shrikhande graph shares these parameters with exactly one other graph, the 4×4 rook's graph, i.e., the line graph L(K4,4) of the complete bipartite graph K4,4
Shrikhande_graph
Graph with tight clique-coloring relation
clique problem, and maximum independent set problem can all be solved in polynomial time, despite their greater complexity for non-perfect graphs. In addition
Perfect_graph
graph is an integral graph if all of the roots of the characteristic polynomial of its adjacency matrix are integers. The notion was introduced in 1974
Integral_graph
Integer sequence
numbers Gregory coefficients Bernoulli polynomials Bernoulli polynomials of the second kind Stirling polynomials Kaneko, Masanobu (1997), "Poly-Bernoulli
Poly-Bernoulli_number
Class of undirected graphs defined from systems of sets
meaning that the induced subgraph of the neighbors of any vertex is a rook's graph. More precisely, in the Johnson graph J ( n , k ) {\displaystyle J(n
Johnson_graph
Mathematical graph of a Sudoku
{\displaystyle Z_{n}^{4}} . The Sudoku graph contains as a subgraph the rook's graph, which is defined in the same way using only the rows and columns
Sudoku_graph
Strategy board game
determine whether a specified player has a winning strategy. And if a polynomial bound is placed on the number of moves that are allowed in between jumps
Checkers
Academic discipline
uses the same pieces as classical chess (8 pawns, 2 knights, 2 bishops, 2 rooks, 1 queen, 1 king) and is won in the same manner (by capturing the opponent's
Quantum_game_theory
Spanish civil engineer (1852–1936)
construction of machines capable of solving real and complex roots of polynomials. At the beginning of the 20th century, he made significant aeronautical
Leonardo_Torres_Quevedo
network flow and other combinatorial optimization problems; identified polynomial-time computability with the intuitive notion of algorithmic efficiency;
List of pioneers in computer science
List_of_pioneers_in_computer_science
German chess player (1868–1941)
algebra, which included proving the primary decomposition of the ideals of polynomial rings. In addition to his better-known works, he also produced philosophical
Emanuel_Lasker
103115. Bloom, Jonathan; Vatter, Vincent (2016), "Two vignettes on full rook placements" (PDF), Australasian Journal of Combinatorics, 64 (1): 77–87,
Enumerations of specific permutation classes
Enumerations_of_specific_permutation_classes
fields of characteristic three. With the 3 × 3 {\displaystyle 3\times 3} Rook's graph and the Games graph, it is one of only three possible strongly regular
Brouwer–Haemers_graph
Russian mathematician
later improved by Kerov to a central limit theorem. Kerov, S.V. (1999). "Rooks on ferrers boards and matrix integrals". Journal of Mathematical Sciences
Sergei_Vasilyevich_Kerov
Difference engine, an automatic mechanical calculator designed to tabulate polynomial functions, in a paper to the Royal Astronomical Society entitled "Note
List of British innovations and discoveries
List_of_British_innovations_and_discoveries
1016/s0008-8846(98)00165-3. Zarandi, MH Fazel; et al. (2008). "Fuzzy polynomial neural networks for approximation of the compressive strength of concrete"
List of datasets for machine-learning research
List_of_datasets_for_machine-learning_research
first computer game ... The machine played a KRK chess endgame, playing rook and king against a person playing a lone king. "Torres Quevedo" (in Spanish)
Science and technology in Spain
Science_and_technology_in_Spain
Graph representing edges of another graph
corresponds to maximum matching in G. Since maximum matchings may be found in polynomial time, so may the maximum independent sets of line graphs, despite the
Line_graph
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