Cylindric Naruse hook-length formula

Let $m,\ell \in \mathbb{Z}_{\geq 1}$ and let $\lambda,\mu \in \mathcal{P}_{m,\ell}$ satisfy $\lambda \supset \mu$. Put

$$n=|\lambda/\mu|=|\mathring{\lambda}/\mathring{\mu}|.$$

Here $f^{\mathring{\lambda}/\mathring{\mu}}$ denotes the number of linear extensions of the cylindric skew diagram, $\mathcal{E}_{\mathring{\lambda}}(\mathring{\mu})$ is the set of cylindric excited diagrams of $\mathring{\mu}$ in $\mathring{\lambda}$, and $h_{\mathring{\lambda}}(x)$ is the hook length of $x$ in $\mathring{\lambda}$. Cylindric Naruse hook-length formula.

$$f^{\mathring{\lambda}/\mathring{\mu}}=n!\sum_{D\in\mathcal{E}_{\mathring{\lambda}}(\mathring{\mu})}\prod_{x\in\mathring{\lambda}\setminus D}\frac{1}{h_{\mathring{\lambda}}(x)}.$$

This is the cylindric analogue of Naruse's hook-length formula for ordinary skew diagrams, expressing the number of linear extensions through cylindric excited diagrams and hook lengths.