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Cayley graph

Let be a group, and let be a set of group elements such that the identity element . The Cayley graph associated with is then defined as the directed graph having one vertex associated with each group element and directed edges whenever . The Cayley graph may depend on the choice of a generating set, and is connected iff generates (i.e., the set are group generators of ).Care is needed since the term "Cayley graph" is also used when is implicitly understood to be a set of generators for the group, in which case the graph is always connected (but in general, still dependent on the choice of generators). This sort of Cayley graph of a group may be computed in the Wolfram Language using CayleyGraph[G], where the generators used are those returned by GroupGenerators[G].To complicate matters further, undirected versions of proper directed Cayley graphs are also known as Cayley graphs.An undirected Cayley graph of a particular generating set of..

Factorial

The factorial is defined for a positive integer as(1)So, for example, . An older notation for the factorial was written (Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978; Conway and Guy 1996).The special case is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).The factorial is implemented in the Wolfram Language as Factorial[n] or n!.The triangular number can be regarded as the additive analog of the factorial . Another relationship between factorials and triangular numbers is given by the identity(2)(K. MacMillan, pers. comm., Jan. 21, 2008).The factorial gives the number of ways in which objects can be permuted. For example, , since the six possible permutations of are , , , , , . The first few factorials for , 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (OEIS A000142).The..

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