Let $w\in S_n$ be a permutation. Is there a reasonable "formula" for the number of elements of the initial interval $[e,w]$ of weak (Bruhat) order from the identity to $w$?

In terms of what such a "formula" might look like: if $w$ is a Grassmannian permutation of shape $\lambda$ then we have $[e,w]\simeq[\varnothing,\lambda]$, an initial interval of Young's lattice, and we can use the determinantal formula of MacMahon mentioned here: Formula for number of edges in Hasse diagram of Young's lattice interval. More generally, if $w$ is a *fully commutative* permutation (i.e., is 321-avoiding), then $[e,w]\simeq [\mu,\lambda]$ for some skew shape $\lambda/\mu$, and we can use the linked formula of Kreweras.

Of course what a formula could look like depends on how we encode $w$, but I would be happy with anything reasonable (e.g., Lehmer code, co-code, etc.).