# Naming conventions for representations of combinations

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?

• I guess a first naming convention here is that these are not combinations, but multisets; combinations typically don't allow for repetition. – Misha Lavrov Jan 27 at 22:55
• I've taken it straight from Wikipedia – davidhigh Jan 27 at 23:19
• Note that Wikipedia avoids referring to these as just "combinations"; of course, if you call something a "combination with repetition", it is unambiguous whether repetition is allowed. – Misha Lavrov Jan 27 at 23:20

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 $$S$$.