A Simple Transformation for Mahonian Statistics on Labelings of Rake Posets

Abstract

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 $$\mathfrak {S}_n$$ of permutations of $$\{1,2,\dots ,n\}$$ . 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 $$\mathfrak {S}_n$$ satisfying the condition that the elements $$1,2,\dots ,k$$ appear in increasing order.