## Estimating the number of functions which are at most $c$-to-$1$ for some constant $c \ge 2$

Notation: $$[m] := \{1, 2, \dots, m \}$$.

How many functions are there $$f: [a] \to [b]$$? The answer is easily seen to be $$b^a$$.

How many $$1$$-to-$$1$$ functions are there $$f: [a] \to [b]$$? Again the answer is well known, and it is sometimes called the falling factorial: $$b(b-1) \dots (b-a+1).$$

How many functions are there $$f: [a] \to [b]$$ that are no more than $$c$$-to-$$1$$?

I don’t expect that there is an exact formula, and I am more interested in the asymptotics. For example, can we give “reasonable” upper and lower bounds, in the case that $$c \ge 2$$ and $$|A| / |B|$$ are fixed, and $$|A| \to \infty$$?

For a concrete example, roughly how many functions are there $$[5n] \to [n]$$ that are at most $$8$$-to-$$1$$? Call this function $$g(n)$$.

Clearly we have $$\frac{(5n)!}{5!^n} \le g(n) \le n^{5n}.$$ The function $${(5n)!}/{5!^n}$$ counts functions that are exactly 5-to-1 (which all satisfy the criterion that they are at most 8-to-1), and the function $$n^{5n}$$ counts all functions.

Applying Stirling’s approximation to the first function gives something like $$\alpha^n n^{5n} \le g(n) \le n^{5n},$$ for some small constant $$\alpha > 0$$.

It seems like there is room for improvement. Is it true, for example, that $$\log g(n) = 5n \log n + C n + o(n)$$ for some constant $$C > 0$$?