Enumeration of measurable functions between finite measurable spaces

Abstract

Let \(X\) and \(Y\) be finite sets with \(|X|=n\), \(|Y|=m\), 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 \(f\colon X\to Y\). 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 \(\exp(\sum_j(e^{b_jx}-1))\). 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 \(N_{\mathcal B}(n)\).