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  2. Permutation representation - Wikipedia

    en.wikipedia.org/wiki/Permutation_representation

    Now the character of this representation is defined as the trace of this permutation matrix. An element on the diagonal of a permutation matrix is 1 if the point in is fixed, and 0 otherwise. So we can conclude that the trace of the permutation matrix is exactly equal to the number of fixed points of .

  3. Permutation - Wikipedia

    en.wikipedia.org/wiki/Permutation

    An exceedance of a permutation σ 1 σ 2...σ n is an index j such that σ j > j. If the inequality is not strict (that is, σ j ≥ j), then j is called a weak exceedance. The number of n-permutations with k exceedances coincides with the number of n-permutations with k descents. [46]

  4. Permutation graph - Wikipedia

    en.wikipedia.org/wiki/Permutation_graph

    Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the modular decomposition. [1]

  5. Cycles and fixed points - Wikipedia

    en.wikipedia.org/wiki/Cycles_and_fixed_points

    (1) There is only one way to construct a permutation of k elements with k cycles: Every cycle must have length 1 so every element must be a fixed point. (2.a) Every cycle of length k may be written as permutation of the number 1 to k; there are k! of these permutations.

  6. Combinations and permutations - Wikipedia

    en.wikipedia.org/wiki/Combinations_and_permutations

    Combinations and permutations in the mathematical sense are described in several articles. Described together, in-depth: Twelvefold way; Explained separately in a more accessible way: Combination; Permutation; For meanings outside of mathematics, please see both words’ disambiguation pages: Combination (disambiguation) Permutation ...

  7. 100 prisoners problem - Wikipedia

    en.wikipedia.org/wiki/100_prisoners_problem

    Graph representations of the permutations (1 7 5)(2 4 8)(3 6) and (1 3 7 4 5 8 2)(6) The prison director's assignment of prisoner numbers to drawers can mathematically be described as a permutation of the numbers 1 to 100.

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