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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.
In combinatorics, stars and bars (also called "sticks and stones", [1] "balls and bars", [2] and "dots and dividers" [3]) is a graphical aid for deriving certain combinatorial theorems. It can be used to solve a variety of counting problems , such as how many ways there are to put n indistinguishable balls into k distinguishable bins. [ 4 ]
For example, flip page effects can be found in the online digital libraries HathiTrust [1] and Internet Archive, [2] and in commercial reading apps such as Paperturn, 3D Issue [3] and Issuu. [4] An early implementation of the effect was the flipping page effect in Macromedia Flash applications in the late 1990s.
The Fano matroid, derived from the Fano plane.Matroids are one of many kinds of objects studied in algebraic combinatorics. Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
For the usual chessboard (8 × 8), G 8 = 2 4 × 4! = 16 × 24 = 384 centrally symmetric arrangements of 8 rooks. One such arrangement is shown in Fig. 2. For odd-sized boards (containing 2n + 1 ranks and 2n + 1 files) there is always a square that does not have its symmetric double − this is the central square of the board. There must always ...
A flip book, flipbook, [1] flicker book, or kineograph is a booklet with a series of images that very gradually change from one page to the next, so that when the pages are viewed in quick succession, the images appear to animate by simulating motion or some other change. Often, flip books are illustrated books for children, but may also be ...
The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology.. In 1978 the situation was reversed—methods from algebraic topology were used to solve a problem in combinatorics—when László Lovász proved the Kneser conjecture, thus beginning the new field of topological combinatorics.
The bipartite double cover of the Petersen graph is the Desargues graph: K 2 × G(5,2) = G(10,3). The bipartite double cover of a complete graph K n is a crown graph (a complete bipartite graph K n,n minus a perfect matching). In particular, the bipartite double cover of the graph of a tetrahedron, K 4, is the graph of a cube.