# Concise notation for “Pairs of all items $\{ x, y, z \}^2$ without $\langle x,x \rangle$, $\langle y,y \rangle$, $\langle z,z \rangle$”

Is there a shorter notation for

Pairs of all items $\{ x, y, z \}^2$ without $\langle x,x \rangle$, $\langle y,y \rangle$, $\langle z,z \rangle$

i.e. given an arbitrary set of items, construct all possible pairs of those items excluding pairs which have the same item on the left and right side?

• My guess is that you care about the order; otherwise, $\binom X2$, as defined in this MO post by Richard Stanley, would be great. – Dylan Moreland Sep 3 '11 at 11:38
• You can try $X^2 \setminus \Delta(X)$. Here $\Delta$ is the diagonal. – Yuval Filmus Sep 3 '11 at 11:40
• I've seen $X^{(n)}$ used in various places for the set of $n$-tuples with pairwise distinct entries. But you'd need to say what it stands for anyway. – t.b. Sep 3 '11 at 12:49
• You could write it as the subset of the power set of $\{x, y, z\}$ containing cardinality 2 elements. This is, I think, the most concise if you're thinking of making long tuples of bigger sets, instead of just pairs or the single triple available. I dunno if there is a standard notation for power set where you care about the order of the elements within the elements, though... – Arthur Sep 3 '11 at 14:50
• Hey Ellie! There is no special notation in mathematics known as "concise notation." Rather, the word "concise" is a very general adjective here that roughly means "expressing a lot of information in very few symbols," or "saying a lot with very little." People like notation that's concise because it's easy and looks nice. (You don't have enough reputation points to post a comment, so I'll flag a moderator to turn it into one for you. If you have any more questions feel free to try again here.) Hope that helps, – anon Jan 27 '12 at 4:46

I don't think you'll get much more concise than either your own proposal or $$\{\, \langle x,y \rangle \in A^2 \mid x \ne y \,\}$$ if you want to be understood without spending ink defining your notation first.
• It depends on the area. The notation $\binom{A}{2}$ is becoming standard in enumerative combinatorics (notwithstanding that it doesn't fit here). – Yuval Filmus Sep 3 '11 at 12:08
• By the same generalization, $A^{\underline{2}}$ or $(A)_2$ (both borrowed from notations for falling factorials) could work for this. They would certainly need an explicit definition in-text, though. – Henning Makholm Sep 3 '11 at 19:27