Enumerative Combinatorics · Exponential generating functions and labeled structures
If aₙ counts structures on a specified set of n distinct labels, its exponential…
Problem
If \(a_n\) counts structures on a specified set of \(n\) distinct labels, its exponential generating function is \(A(x)=\sum_{n\ge0}a_nx^n/n!\). Explain how to recover \(a_n\) from \(A\) and find the EGF when each label independently receives one of three colors. Prove that multiplying EGFs \(A(x)\) and \(B(x)\) counts an ordered pair of structures placed on disjoint label sets whose union is the full label set. Apply this to a permutation on one chosen subset and an unstructured set of the remaining labels, allowing either subset to be empty. Find the resulting EGF and the count on four labels.
Hint
Compare the EGF coefficient convention with the number of structures being counted.
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