How to calculate combinations count for this problem I will explain my question using simple example, cause I don't know to descrive it properly.
If we have 2 numbers $\{a,b\}$, by comparing them, we get 3 possible combinations: $$a&gt;b, \hspace{3pt} a&lt;b, \hspace{3pt} a=b$$ For 3 numbers $\{a,b,c\}$ we get, $$b&lt;a&gt;c, \hspace{3pt} b&gt;a&lt;c, \hspace{3pt} a&lt;b&gt;c, \hspace{3pt} a&gt;c&lt;b, \hspace{3pt} \hspace{3pt} a=b=c, \hspace{3pt} a=b&gt;c, \hspace{3pt} etc...$$

How to calculate how many combinations exist for n numbers.
If I understand the problem correctly, these are the ordered Bell numbers, aka the Fubini numbers. Perhaps the cleanest formulation of the problem is that we have $n$ (distinct) runners in a race, and we want to count the number $a(n)$ of possible orders of finish, including ties. 

There are useful recurrences, ways to express $a(n)$ as sums,  and asymptotics, but no known closed form.
Put the n numbers in any order. There are $(n-1)$ spaces between the numbers in which you have to put signs. There are 3 signs to choose from for each space. Therefore, the solution is 3^(n-1).