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$\ D_n = \sum_{i,j,k}^n ε_{ijk}···a_i b_j c_k ··· , $

where $\ ε_{ijk···} $, analogous to the Levi-Civita symbol of Section 2.9, is +1 for even permutations1 (ijk ···) of (123 ···n),−1 for odd permutations, and zero if any index is repeated.

This is from Arfken's Mathematical methods for Physicists 6th edition. Now I understand that 123 has 6 permutations. But what are odd and even permutations?!

mrateb
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    As for in the case of $S_3$, the odd permutations are $(12),(13),(23)$ (or alternatively notated 213, 321, 132), and the even permutations are $(1), (12)(23)$ and $(13)(32)$ (or alternately notated: 123, 231, 312) – JMoravitz Nov 20 '19 at 18:01
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    In short, every permutation can be expressed as a finite sequence of exchanges of pairs. In fact, there are infinitely many such sequences that result in the given permutation. For some permutations, the lengths of those sequences will aways be even. For all the remaining permutations, the lengths of those sequences will always be odd. There is no permutation that is representable by both a sequence of even length and a sequence of odd length. Thus we define the parity of the permutation to be the parity of the sequences of exchanges that produce it. – Paul Sinclair Nov 21 '19 at 01:17

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