The sorting index can also be described as the total distance the elements in [sigma] travel when [sigma] is sorted using the Straight Selection Sort algorithm  in which, using a transposition, we move the largest number to its proper place, then the second largest to its proper place, etc.
The relationship to other Mahonian statistics and the Eulerian partner for sor were studied by Wilson  who called the sorting index DIS.
To obtain the results, in Section 2 we define a sorting index and cycles for perfect matchings and study the distributions of these statistics over matchings of fixed type.
Analogously to sor, Petersen defined the sorting index for signed permutations of type [B.
While space constraints prevent us from providing details in this extended abstract, we mention that in  we define a sorting index and cycle number for bicolored matchings in a fashion analogous to what we will show for ordinary matchings.
n](D), we define the sorting index of M with respect to M0, denoted by sor(M, [M.
0] [member of] M(D(r)) the sorting index sor(*, [M.