Odd Permutation

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

For a set of n elements and n>=2, there are n!/2 odd permutations (D'Angelo and West 2000, p. 111), which is the same as the number of even 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, Even Permutation, Permutation

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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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Odd Permutation

Cite this as:

Weisstein, Eric W. "Odd Permutation." From MathWorld--A Wolfram Web Resource.

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