Naming conventions for representations of combinations

Question

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?

Answer 1

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 $S$.