|
Ménage Problem
In combinatorics, combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so that men and women alternate and nobody sits next to his or her partner. (''Ménage'' is the French language, French word for "household", referring here to a male-female couple.) This problem was formulated in 1891 by Édouard Lucas and independently, a few years earlier, by Peter Guthrie Tait in connection with knot theory. For a number of couples equal to 3, 4, 5, ... the number of seating arrangements is :12, 96, 3120, 115200, 5836320, 382072320, 31488549120, ... . Mathematicians have developed formulas and recurrence equations for computing these numbers and related sequences of numbers. Along with their applications to etiquette and knot theory, these numbers also have a graph theory, graph theoretic interpretation: they count the numbers of matching (graph theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wedding Banquet Setting
A wedding is a ceremony in which two people are united in marriage. Wedding traditions and customs vary greatly between cultures, ethnicity, ethnicities, Race (human categorization), races, religions, Religious denomination, denominations, Country, countries, social classes, and sexual orientations. Most wedding ceremonies involve an exchange of marriage vows by a couple; a presentation of a gift (e.g., an offering, rings, a symbolic item, flowers, money, or a dress); and a public proclamation of marriage by an authority figure or Celebrant (Australia), celebrant. Special wedding garments are often worn, and the ceremony is sometimes followed by a wedding reception. Music, poetry, prayers, or readings from religious texts or literature are also commonly incorporated into the ceremony, as well as Wedding superstitions, superstitious customs. Common elements across cultures Some cultures have adopted the traditional Western custom of the white wedding, in which a bride wear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inclusion–exclusion Principle
In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union (set theory), union of two finite sets; symbolically expressed as : , A \cup B, = , A, + , B, - , A \cap B, where ''A'' and ''B'' are two finite sets and , ''S'', indicates the cardinality of a set ''S'' (which may be considered as the number of elements of the set, if the set is Finite set, finite). The formula expresses the fact that the sum of the sizes of the two sets may be too large since some elements may be counted twice. The double-counted elements are those in the intersection (set theory), intersection of the two sets and the count is corrected by subtracting the size of the intersection. The inclusion-exclusion principle, being a generalization of the two-set case, is perhaps more clearly seen in the case of three sets, which for the sets ''A'', ''B'' and ''C'' is given by :, A \cup B \cup C, = , A, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot (mathematics)
In mathematics, a knot is an embedding of the circle () into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term ''knot'' is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory. Formal definition A knot is an embedding of the circle () into three-dimensional Euclidean space (), or the 3-sphere (), since the 3-sphere is compact. Two knots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circulant Matrix
In linear algebra, a circulant matrix is a square matrix in which all rows are composed of the same elements and each row is rotated one element to the right relative to the preceding row. It is a particular kind of Toeplitz matrix. In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform. They can be interpreted analytically as the integral kernel of a convolution operator on the cyclic group C_n and hence frequently appear in formal descriptions of spatially invariant linear operations. This property is also critical in modern software defined radios, which utilize Orthogonal Frequency Division Multiplexing to spread the symbols (bits) using a cyclic prefix. This enables the channel to be represented by a circulant matrix, simplifying channel equalization in the frequency domain. In cryptography, a circulant matrix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
0-1 Matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. It is an important tool in combinatorial mathematics and theoretical computer science. Matrix representation of a relation If ''R'' is a binary relation between the finite indexed sets ''X'' and ''Y'' (so ), then ''R'' can be represented by the logical matrix ''M'' whose row and column indices index the elements of ''X'' and ''Y'', respectively, such that the entries of ''M'' are defined by :m_ = \begin 1 & (x_i, y_j) \in R, \\ 0 & (x_i, y_j) \not\in R. \end In order to designate the row and column numbers of the matrix, the sets ''X'' and ''Y'' are indexed with positive integers: ''i'' ranges from 1 to the cardinality (size) of ''X'', and ''j'' ranges from 1 to the cardinality of ''Y''. See the article on indexed sets for more detail. The transpose R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permanent (mathematics)
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant. Definition The permanent of an matrix is defined as \operatorname(A)=\sum_\prod_^n a_. The sum here extends over all elements σ of the symmetric group ''S''''n''; i.e. over all permutations of the numbers 1, 2, ..., ''n''. For example, \operatorname\begina&b \\ c&d\end=ad+bc, and \operatorname\begina&b&c \\ d&e&f \\ g&h&i \end=aei + bfg + cdh + ceg + bdi + afh. The definition of the permanent of ''A'' differs from that of the determinant of ''A'' in that the signatures of the permutations are not taken into account. The permanent of a matrix A is denoted per ''A'', perm ''A'', or Per ''A'', sometimes with parentheses around the argument. Minc uses Per(''A'') for the permanent of rectangula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alice's Adventures In Wonderland
''Alice's Adventures in Wonderland'' (also known as ''Alice in Wonderland'') is an 1865 English Children's literature, children's novel by Lewis Carroll, a mathematics university don, don at the University of Oxford. It details the story of a girl named Alice (Alice's Adventures in Wonderland), Alice who falls through a rabbit hole into a fantasy world of anthropomorphism, anthropomorphic creatures. It is seen as an example of the literary nonsense genre. The artist John Tenniel provided 42 wood-engraved illustrations for the book. It received positive reviews upon release and is now one of the best-known works of Victorian literature; its narrative, structure, characters and imagery have had a widespread influence on popular culture and literature, especially in the fantasy genre. It is credited as helping end an era of didacticism in children's literature, inaugurating an era in which writing for children aimed to "delight or entertain". The tale plays with logic, giving th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Matching
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph with edges and vertices , a perfect matching in is a subset of , such that every vertex in is adjacent to exactly one edge in . The adjacency matrix of a perfect matching is a symmetric permutation matrix. A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs:Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985, Chapter 5. : In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crown Graph
In graph theory, a branch of mathematics, a crown graph on vertices is an undirected graph with two sets of vertices and and with an edge from to whenever . The crown graph can be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed, as the bipartite double cover of a complete graph, as the tensor product , as the complement of the Cartesian direct product of and , or as a bipartite Kneser graph representing the 1-item and -item subsets of an -item set, with an edge between two subsets whenever one is contained in the other. Examples The 6-vertex crown graph forms a cycle, and the 8-vertex crown graph is isomorphic to the graph of a cube. In the Schläfli double six, a configuration of 12 lines and 30 points in three-dimensional space, the twelve lines intersect each other in the pattern of a 12-vertex crown graph. Properties The number of edges in a crown graph is the pronic number . Its achromatic number is : one ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
