A Simple Transformation for Mahonian Statistics on Labelings of Rake Posets

We present a simple transformation for the inversion number and major index statistics on the labelings of a rooted tree with n vertices in the form of a rake with k teeth. The special case k= 0 provides a simple transformation for the Mahonian statistics on the set S_n of permutations of { 1 , 2 , ⋯ , n}. We also extend the transformation to a bijective interpretation of the fact that the major index of the equivalence classes of the labelings is equidistributed with the major index of the permutations in S_n satisfying the condition that the elements 1 , 2 , ⋯ , k appear in increasing order.