有限可测空间之间可测函数的计数
Enumeration of measurable functions between finite measurable spaces
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中文总结 AI 辅助
该研究计数有限可测空间间的可测函数，推导了离散与一般σ代数下的计数公式，关联完全Bell多项式与复合泊松变量，还探讨了计数结果的渐近增长特性。
中文摘要 AI 辅助
设X和Y是基数分别为|X|=n、|Y|=m的有限集，分别配备σ代数A和B。对X上的任意σ代数A及Y上的任意σ代数B，我们对可测函数f:X→Y进行计数。当B是离散σ代数时，有序对(A,f)的数量为Touchard多项式Tₙ(m)=∑ₖS(n,k)mᵏ；当B为一般σ代数且原子大小为b₁,…,bᵣ时，X上所有σ代数A对应的有序对(A,f)的数量是幂和pₐ=∑ⱼbⱼᵃ下的完全Bell多项式N_B(n)，其指数生成函数为exp(∑ⱼ(e^bʲˣ−1))，该式在离散情形下退化为Touchard多项式，且在平凡陪域σ代数时取最大值。我们进一步证明N_B(n)等于复合泊松随机变量Z的n阶矩E[Zⁿ]，并讨论N_B(n)的基本渐近增长特性。
英文摘要
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 \(N_{\mathcal B}(n)\).
