Consider two countable families of objects, given as unions of finite subfamilies:

$F^k = \bigcup\_{n \in \mathbb{N}} F^k\_n$, *k* = 1,2.

Let there be a bijection $f: F^1 \rightarrow F^2$ such that $x \in F^1\_n \Leftrightarrow f(x) \in F^2\_n$ for all *n* $\in \mathbb{N}$.

This means: The two families have the same counting functions $f^1(n) = f^2(n)$ for all *n* $\in \mathbb{N}$ with $f^k(n) = |F^k\_n|$.

This may be *by sheer accident*, or it may be because the two families are in some sense *essentially the same*.

> 
> Can the notions of "by accident" and
>  "essentially the same" be distinguished in this context, and how?
> 
> 
> 

"Essentially the same" might mean: "there is a *computable* bijection" and "by accident" might mean: "there is *no* computable bijection". Or might category theoretical notions lead further?

> 
> Are there known examples of two
>  families as above that have the same counting
>  functions by accident (in the meaning just mentioned or another one)?
> 
> 
> 

PS: More sensible tags are welcome!