You can search this statistic (normalized so that the smallest value is 0) in www.FindStat.org and you will find that this is the rank of the permutation inside the lattice of alternating sign matrices. You find further information at http://www.findstat.org/St000055.

You can search this statistic (normalized so that the smallest value is 0) in www.FindStat.org and you will find that this is the rank of the permutation inside the lattice of alternating sign matrices. You find further information at http://www.findstat.org/St000055 and the references there:

Sack, J., Úlfarsson, H. Refined inversion statistics on permutations MathSciNet:2880660

Striker, J. A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux MathSciNet:2794039