The second Fuss-Catalan number of a finite Cartan type. The Fuss-Catalan numbers of a finite Cartan type are given by $$\frac{1}{|W|}\prod (d_i+mh) = \prod \frac{d_i+mh}{d_i}$$ where the products run over all degrees of homoneneous fundamenal invariants of the Weyl group of a Cartan type.

The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.