Problem: What is the formula of number of arrangements? More specifically I need to avoid repeated elements and the order of the sequence does not matter. For lucidity I show an example:
For 3 elements:
1
2
3
1 2
1 3
2 4
1 2 3
For 4 elements:
1
2
3
4
1 2
1 3
1 4
2 3
1 2 3
1 2 4
1 3 4
2 3 4
1 2 3 4
I really don't know the mathematical name of this arranging, I suppose the form of this is *((n-1)n)+1. So what is this exactly?
You are asking for the number of non-empty subsets of a set. If the set has $n$ elements, it has $2^n-1$ non-empty subsets.
Imagine that the elements of the set are lined up in a row. To make a subset, we say "yes" or "no" to each of the elements. There are $2^n$ ways to do this, but one of them (all no) produces the empty set.
The number of $k$ element subsets pf a set of $m$ elements is often denoted by $\binom{n}{k}$. It turns out that $\binom{n}{k}=\frac{n!}{k!(n-k)!}$.
Remark: So for example the set $\{1,2,3,4\}$ has $2^4-1=15$ non-empty subsets. The list in the OP has $13$. because $\{2,4\}$ and $\{3,4\}$ are missing.