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?

  • $\begingroup$ I guess a first naming convention here is that these are not combinations, but multisets; combinations typically don't allow for repetition. $\endgroup$ – Misha Lavrov Jan 27 at 22:55
  • $\begingroup$ I've taken it straight from Wikipedia $\endgroup$ – davidhigh Jan 27 at 23:19
  • $\begingroup$ 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. $\endgroup$ – 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 $S$.

  • $\begingroup$ Thanks a lot. Seems like I have to stick to the quantum mechanics terminology. $\endgroup$ – davidhigh Jan 27 at 23:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.