combinatorics (kŏmˌbənətôrˈĭks) [key] or combinatorial analysis kŏmˌbĭnətôrˈēəl, sometimes called the science of counting, the branch of mathematics concerned with the selection, arrangement, and operation of elements within sets. Combinatorial theory deals with existence (does a particular arrangement exist?), enumeration (how many such arrangements are there?) and structure (what are the properties of each arrangement?). It has applications in such diverse areas as managing computer and telecommunication networks, predicting poker hands, dividing tasks among workers, and finding a pair of socks in a drawer. Because combinatorics deals with concrete problems by limiting itself to finite collections of discrete objects, as opposed to the more common, continuous mathematics, it has neither standard algebraic manipulations nor a systematic problem-solving framework. Instead it relies upon the logical analysis of possibilities for each new problem, breaking the problem into a series of steps and substeps. Combinatorics has its roots in the 17th- and 18th-century attempts to analyze the odds of winning at games of chance. The advent of computers in the 20th cent. made possible the high-speed calculation required to analyze the multitude of possibilities inherent in a combinatorial approach to large-scale statistical testing and analysis. Branches of combinatorics include graph theory and combinations and permutations.
See A. Slomson, An Introduction to Combinatorics (1991); A. Tucker, Applied Combinatorics (3d ed. 1994); R. A. Brualdi, Introductory Combinatorics (3d ed. 1997); M. Hall, Combinatorial Theory (2d ed. 1998); R. P. Stanley, Enumerative Combinatorics (1999).
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