If by "parity of the permutation" you mean the number of elements out of place (or whether that number is even or odd, or the sum of the counts of elements that are on the wrong side of an element over all elements), then I think it's not possible. As Perl uses quicksort as its sorting algorithm, and that sorting algorithm swaps elements across large distances, you can't keep track of the parity in its callback.

You can maybe look at the implementation and infer that Perl will always call the sort block with $a being the left element and $b being the right element in the sequence. But then determining a parity from the number of calls your callback receives and the elements that get compared still strikes me as fragile if even possible at all.

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