A weighted enumeration of maximal chains in the Bruhat order (Q1604772)

Given a finite Weyl group \(W\) with root system \(\Phi\), assign the weight \(\alpha\in\Phi\) to each covering pair in the Bruhat order related by the reflection corresponding to \(\alpha\). Extending this multiplicatively to chains, the author shows that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula: \[ \sum\text{wt}(x_0<\cdots<x_l)=|\Phi|!\prod_{\alpha\in\Phi}\alpha/\text{ht}(\alpha), \] where the sum ranges over all maximal-length chains in the Bruhat ordering of \(W\). Moreover, he proves a similar result for a weighted sum over maximal chains in the Bruhat ordering of any parabolic quotient of \(W\): \[ \sum\text{wt}_\lambda(\mu_0<\cdots<\mu_l)=|\Phi_\lambda|!\prod_{\alpha\in\Phi_\lambda}\langle\lambda,\alpha\rangle/\text{ht}(\alpha), \] where \(\lambda\in{\mathbf E}\) is dominant and \(\Phi_\lambda:=\{\alpha\in\Phi\mid\langle\lambda,\alpha\rangle>0\}\), and the sum ranges over all maximal-length chains in the Bruhat ordering of \((W\lambda,<)\).