Is there a name for cartesian product excluding commutative duplicates?

Question

Lots of algorithms have a pair of nested for loops where we need to consider elements belonging to the set

$i,j\in\mathbb{N}, \ 1 \le i \le j$

(Though in practice we often only consider the natural numbers up to some certain upper bound $n$)

Is there a name for this? It's similar to the cartesian product but excludes "duplicates" (where $(i,j) = (j,i)$)

Answer 1

I guess this was a daft question - must've been tired! -it's just the set of 2-combinations from the set $\{1, 2, ... , n\}$

Answer 2

Given $S \times S$, equip it with the action of $\mathbb Z_2$ that permutes the elements, and mod out by this action, $(S \times S)/\mathbb Z_2$. When $S$ has further structure (such as a module or vector space one) we sometimes call this $S^2(V)$, the second symmetric power of $V$.