Multivariate $P$- and $Q$-polynomial structures of generalized Johnson association schemes

Let $\mathfrak X$ be a commutative association scheme of class $d$ on $X$. We construct the generalized Johnson association scheme $\mathfrak J_{n,h}(\mathfrak X)$ on partial maps from $[n]$ to $X$ with domains of size $h$. For $1\le h\le n-1$, we prove that $\mathfrak J_{n,h}(\mathfrak X)$ has $(d+