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.