MajorIndex on DyckPath
The sum of the descent positions of the step word — where an up step is followed by a down step.
Same name as the permutation statistic, genuinely a different function — which is why definitions are keyed by signature.
Called as CombinatorialStat(x, "MajorIndex"). The head's page has its examples.
Also on
The same name on other carriers, each its own implementation: Permutation.