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**Unformatted text preview: **to 312, and two is even, we’ll say that 312 is an even permutation. Since 132 only took one
transposition, we’ll call 132 an odd permutation.
Note that a transposition takes any two numbers in the list, and they switch places. We could restrict
ourselves to only transposing adjacent numbers in the list, since any transposition is a combination of an
odd number of adjacent transpositions. For example, consider the list
(9) 12345, and I want to transpose the 2 and 5. It’ll take three adjacent transpositions to get the 2 to the end
(10) 12345 → 13245 → 13425 → 13452, and then two more adjacent transpositions to the 5 into the second spot
(11) 13452 → 13542 → 15342. Note that both the 2 and 5 have to jump over the 3 and 4, that’s even. And then the 5 and 2 have to switch
places, which makes it odd.
There are an inﬁnite number of ways to get from one permutation to ano...

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