Let $\ \mathbf N = \{1\ 2\ \ldots\}\ $ be the set of natural numbers. Let $\ f : \mathbf N\rightarrow\mathbf N\ $ be an arbitrary function, and $\ \forall_{n\in\mathbf N}\, F(n)\ :=\ \max_{k = 1\ldots n}\, f(k)$.

Let's assume that, with respect to a fixed universal Turing machine, there exists at least one algorithm which computes $\ f,\ $ and let $\ ||f||_A(n)\ $ be the number of Turing operations which compute $\ f(n)\ $ by algorithm $\ A$.

By a polynomial $\ \mathbf N\rightarrow\mathbf N\ $ I mean a function which differs from a (true real) polynomial by less then 1 for almost all natural numbers $\ n\in\mathbf N$.

**DEFINITION 1** Function $\ f\ $ is called a *fast counter* $\ \Leftarrow:\Rightarrow\ $ there exists an algorithm $\ A\ $ and a polynomial $\ p : \mathbf N\rightarrow \mathbf N\ $ such that

$$\forall_{n\in\mathbf N}\ \ ||f||_A(n)\ \le\ \frac{p(n)}{n!}\cdot F(n) $$

**DEFINITION 2** Function $\ f\ $ is called a *slow counter* $\ \Leftarrow:\Rightarrow\ $ for every algorithm $\ A\ $ there exists polynomial $\ q : \mathbf N\rightarrow \mathbf N\ $

$$\forall_{n\in\mathbf N}\ \ ||f||_A(n)\ \ge\ \frac{F(n)}{n!\cdot q(n)}$$

**DEFINITION 3** Function $\ f\ $ is called an *algorithmic counter* $\ \Leftarrow:\Rightarrow\ \ f\ $ is both a fast and a slow counter.

QUESTIONLet $\ pos(n)\ $ be the number of all partial orders in the integer interval $\ \{0\ \ldots\ n\!-\!1\}.\ $ Is function $\ pos\ $ an algorithmic counter?

A similar question holds for the number of quasi-orders (i.e. of topologies).