Major index

In mathematics, the major index of the permutation π is defined as

${\textrm {maj}}(\pi )=\sum _{\pi (i)>\pi (i+1)}i.$

In other words, the major index of π is the sum of the positions of its “descent tops.” For example, the major index of the permutation 25143 (in one-line notation) is 2 + 4 = 6.

This statistic is named after Major Percy Alexander MacMahon who showed in 1913 that the distribution of the major index on all permutations of a fixed length is the same as the distribution of inversions.

References

• MacMahon, P.A. (1913), "The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects", Amer. J. Math., 35: 281–322.