Combinations, i.e. ordered $k$-subsets of the set $\{1,2, \ldots, N\}$, are usually expressed by stating the indices $(i_1,\ldots,i_k)$ contained in a given subset. For example, by choosing $N=3$ and $k=2$, and allowing for repetitions, one obtains (in co-lex order):
$$ (1,1),(1,2),(2,2),(1,3),(2,3),(3,3)\\ $$
The same information can also be represented by a vector $(o_1,\ldots,o_N)$, where the number $o_j = \sum_{m=1}^k \delta_{i_m,j}$ count the number of times the index $j$ appears in the combination. In this representation, one obtains for the previous example:
$$ (2,0,0),(1,1,0),(0,2,0),(1,0,1),(0,1,1),(0,0,2) $$
In quantum mechanics, the first representation could be called orbital representation, whereas the second is called occupation representation.
What are the naming conventions used in combinations?
If you are looking at subsets $S \subseteq \{1,2,\dots,n\}$, the vector $(x_1,x_2, \dots,x_n)$ where $x_i = 1$ if $i\in S$ and $0$ otherwise is called the indicator vector, characteristic vector, or incidence vector.
In your case, we are looking at multisets taken from $\{1,2,\dots,n\}$. In this case, we can borrow the term "characteristic vector" but probably shouldn't use either of the others, since they connote a vector with entries in $\{0,1\}$. For this reason, the indicator vector is sometimes written $1_S$, which is notation you probably shouldn't borrow. Characteristic anythings of $S$ are sometimes denoted $\chi_S$.
Usually, we think of a set $S = \{1,1,2,4\}$ as being the multiset, and the corresponding characteristic vector $(2,1,0,1)$ (if $n=4$) as a representation of the multiset, so there's not much specific terminology for your first representation. In principle, you could call $(1,1,2,4)$ the ordered sequence of elements of $S$.