Geodesic
Odd and Even Permutations
Canadian mathematical bulletin, Tome 3 (1960) no. 2, pp. 185-186

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This note gives a proof for the familiar elementary theorem that if a permutation of the integers 1, ..., n (with n ≥ 2) is expressed as a product π1 of N1 transpositions and also as a product π2 of N2 transpositions, then N1 and N2 are both even or both odd (equivalently: N1 + N2 is even).Here, a transposition means an interchange of two of the integers. If the interchange is between adjacent integers it is called an adjacent-trans position. 
Halperin, Israel. Odd and Even Permutations. Canadian mathematical bulletin, Tome 3 (1960) no. 2, pp. 185-186. doi: 10.4153/CMB-1960-023-1
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