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I read the definition of an even permutation on wiki to be the number of elements of the original sequence had to be exchanged to get the new sequence. By that definition if we start with (1,2,3) then 1,3,2 is an odd permutation because we just exchanged 3 and 2 whereas 3,1, 2 is an even permutation because it involves two exchanges (first 2 and 3 are exchanged and then 3 and 1 are exchanged.

In Springer's Matrix Algebra for Statistics they say: "Define a permutation to be even or odd according to the number of times that a smaller element follows a larger one in the permutation. (For example, 1, 3, 2 is an odd permutation, and 3, 1, 2 is an even permutation.)"

How does this definition square with the one above? For (1,3,2) a smaller value follows a larger one when going from 3 to 2, but not when going from 1 to 3. In (3,1,2) a smaller value follows a larger one when going from 3 to 1, but not when going from 1 to 2. Thus it seems that both permutations have the same 'number of times that a smaller element follows a larger one in the permutation'.

MHall
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The second definition states "the number of times that a smaller element follows a larger one"; it does not specify consecutively. So in the case (3,1,2), we count two such events: 1 following 3 and 2 following 3. It is not hard to show the definitions are equivalent; thinking in positional notation, the exchange itself changes the number of times a smaller number follows a larger one by plus/minus one. The only other changes occur with elements that are both numerically and positionally between the exchanged elements, and those change this value by plus/minus two each.

Jeremy Dover
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