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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 .
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 ...
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. [ 1 ] : 26 An n × n permutation matrix can represent a permutation of n elements.
In mathematics, the Robinson–Schensted correspondence is a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape. It has various descriptions, all of which are of algorithmic nature, it has many remarkable properties, and it has applications in combinatorics and other areas such as representation theory.
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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]
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