ReflectionLength on Permutation

n minus the number of cycles — the minimum number of transpositions.

On Permutation

Called as CombinatorialStat(x, "ReflectionLength"). The head's page has its examples.

Live

FindStat St000216

The absolute length of a permutation.

Object	Value
[1,2]	0
[2,1]	1
[1,2,3]	0
[1,3,2]	1
[2,1,3]	1
[2,3,1]	2
[3,1,2]	2
[3,2,1]	1

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000216 has the rest.