A poset view of the major index

We introduce the Major MacMahon map from Z〈a,b〉Z〈a,b〉 to Z[q]Z[q], and show how this map interacts with the pyramid and bipyramid operators. When the Major MacMahon map is applied to the abab-index of a simplicial poset, it yields the q-analogue of n! times the h-polynomial of the poset. Applying the map to the Boolean algebra gives the distribution of the major index on the symmetric group, a seminal result due to MacMahon. Similarly, when applied to the cross-polytope we obtain the distribution of one of the major indexes on signed permutations due to Reiner.