A descent variant minus the number of inversions.
This statistic is defined for general finite crystallographic root system $\Phi$ with Weyl group $W$ as follows: Let $2\rho = \sum_{\beta \in \Phi^+} \beta = \sum_{\alpha\in\Delta}b_\alpha \alpha$ be the sum of the positive roots expressed in the simple roots.
For $w \in W$ this statistic is then
$$\operatorname{stat}(w) = \sum_{\alpha\in\Delta\,:\,w(\alpha) \in \Phi^-}b_\alpha - \ell(w)\,,$$
where the sum ranges over all descents of $w$ and $\ell(w)$ is the Coxeter length.
It was shown in [1], that for irreducible groups, it holds that
$$\sum_{w\in W} q^{\operatorname{stat}(w)} = f\prod_{\alpha \in \Delta} \frac{1-q^{b_\alpha}}{1-q^{e_\alpha}}\,,$$
where $\{ e_\alpha \mid \alpha \in \Delta\}$ are the exponents of the group and $f$ is its index of connection, i.e., the index of the root lattice inside the weight lattice.
For a permutation $\sigma \in S_n$, this becomes
$$\operatorname{stat}(\sigma) = \sum_{i \in \operatorname{Des}(\sigma)}i\cdot(n-i) - \operatorname{inv}(\sigma)\,.$$

[1] Stembridge, J. R., Waugh, D. J. A Weyl group generating function that ought to be better known MathSciNet:1692145

Aug 08, 2023 at 16:30 by Christian Stump

Aug 08, 2023 at 16:30 by Christian Stump