The r-major index

In this post and this one too, we discussed the inv and maj statistics on words, and two different proofs of their equidistribution. In fact, there is an even more unifying picture behind these statistics: they are simply two different instances of an entire parameterized family of statistics, called $r\text{-maj}$, all of which are equidistributed!

Rawlings defined an $r$-inversion of a permutation $\pi$ to be a pair of entries $(\pi_i,\pi_j)$ with $i\lt j$ and $$0\lt \pi_i-\pi_j\lt r.$$ For instance, $21534$ has three total inversions, $(2,1)$, $(5,4)$, and $(5,3)$, but only the first two have $\pi_i-\pi_j<2$, so it has two $2$-inversions. He also defined an $r$-descent to be an index $i$ for which $$\pi_i\ge \pi_{i+1}+r,$$ so that $21534$ has only position $3$ as a $2$-descent.

Finally, he defines the $r\text{-maj}$ of a permutation to be $$r\text{-}\mathrm{maj}(\pi)=\left(\sum_{\pi_i\ge \pi_{i+1}+r}i\right)+\#r\text{-}\mathrm{inversions}.$$ Thus $2\text{-}\mathrm{maj}(21534)=3+2=5$. Notice that $1\text{-maj}$ is the usual major index, and $n\text{-maj}$ is the inv statistic!

Rawling’s result is that these statistics all have the same distribution: for any $r,s\ge 1$, the number of permutations of $\{1,2,\ldots,n\}$ having $r\text{-maj}$ value $k$ is the same as the number of them having $s\text{-maj}$ value $k$ for any $k$. More succinctly, $$\sum_{\pi\in S_n} q^{r\text{-maj}(\pi)}=\sum_{\pi\in S_n} q^{s\text{-maj}(\pi)}.$$

A colleague of mine mentioned this result and challenged me to prove it without reading the proof first, so here goes. I challenge the reader to do the same before turning to the next page. Good luck!

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