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Even Permutation


An even permutation is a permutation obtainable from an even number of two-element swaps, i.e., a permutation with permutation symbol equal to +1. For initial set {1,2,3,4}, the twelve even permutations are those with zero swaps: ({1,2,3,4}); and those with two swaps: ({1,3,4,2}, {1,4,2,3}, {2,1,4,3}, {2,3,1,4}, {2,4,3,1}, {3,1,2,4}, {3,2,4,1}, {3,4,1,2}, {4,1,3,2}, {4,2,1,3}, {4,3,2,1}).

For a set of n elements and n>2, there are n!/2 even permutations, which is the same as the number of odd permutations. For n=1, 2, ..., the numbers are given by 0, 1, 3, 12, 60, 360, 2520, 20160, 181440, ... (OEIS A001710).


See also

Alon-Tarsi Conjecture, Alternating Group, Odd Permutation, Permutation

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References

D'Angelo, J. P. and West, D. B. Mathematical Thinking: Problem-Solving and Proofs, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.Sloane, N. J. A. Sequence A001710/M2933 in "The On-Line Encyclopedia of Integer Sequences."

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Even Permutation

Cite this as:

Weisstein, Eric W. "Even Permutation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EvenPermutation.html

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