Solution to Permutation (Transposition) Problem

 Express p =

 [1 2 3 4]

 [2 3 1 4] 

 as the product of transpositions, and determine the sign (+1 or -1) of the resulting end permutation. 

Solution:

Let T1 be the transposition 2 <-> 1 leaving 3, 4 fixed, so:

T1 p =

[1 2 3 4]
[1 3 2 4]


Let T2 be the transposition 2 <-> 3 leaving 1, 4 fixed, so:

T2 T1 p =

[1 2 3 4]
[1 2 3 4]

Then:

T2 T1 p = I (identity)

TWO transpositions (T1, T2) operated on p, so that the sign of the resulting permutation (to reach identity) is +1.  The permutation is therefore even.