Enumeration of measurable functions between finite measurable spaces

Let $X$ and $Y$ be finite sets with $\mathrm{\mid }X\mathrm{\mid }=n$, $\mathrm{\mid }Y\mathrm{\mid }=m$, equipped with sigma algebras $\mathcal{A}$ and $\mathcal{B}$. For arbitrary sigma algebras $\mathcal{A}$ on $X$ and $\mathcal{B}$ on $Y$, we enumerate measurable functions $f\text{\hspace{0.17em}⁣}:X\to Y$. When $\mathcal{B}$ is discrete, the number of pairs $(\mathcal{A},f)$ is the Touchard polynomial $T_n(m)={\sum }_{k}S(n,k)m^k$. For general $\mathcal{B}$ with atom sizes $b_1,\dots ,b_r$, the number of pairs $(\mathcal{A},f)$ over all sigma algebras $\mathcal{A}$ on $X$ is the complete Bell polynomial $N_{\mathcal{B}}(n)$ in the power sums $p_a={\sum }_j{b}_j^a$, with exponential generating function $\exp ({\sum }_j(e^{b_{j}x}-1))$. This specialises to the Touchard polynomial in the discrete case and is maximised by the trivial codomain sigma algebra. We further show that $N_{\mathcal{B}}(n)=\mathbb{E}[Z^n]$ for a compound Poisson random variable $Z$, and we discuss basic asymptotic growth of $N_{\mathcal{B}}(n)$.