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I'm kind of confused how to solve this problem:

Find the exponential generating series for the number of functions $f: A → B$ where $A = \{1, 2, 3 ,\dots, n\}$ and $B = \{1, 2, 3, \dots, m\} $ such that the preimage set of every element of $B$ is at least size four?

i.e. # $f^{-1}(j) \geq 4$ for each $j \in B$

Robert Z
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memesnotdreams
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1 Answers1

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Hint. Read the examples (3) and (4) in the document exponential.

Example (3) is about the exponential generating function for the number of functions from $A$ to $B$ (with $|A|=n$ and $|B|=m$). It means that the preimage set of every element of $B$ has at least size ZERO. Such e.g.f. is $$F(x)=(e^x)^m.$$

Example (4) is about the exponential generating function for the number of functions from $A$ to $B$ which are surjective. It means that the preimage set of every element of $B$ has at least size ONE. Such e.g.f. is $$F(x)=(e^x-1)^m.$$

Moreover, the question The total number of ways to put $N$ distinct balls into $K$ distinct boxes so that every box has more than one ball (assuming that $N \geqslant 2K$) is related to the case when the preimage set of every element of $B$ has at least size TWO.

Can you take it from here?

Robert Z
  • 147,345