# 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.

(I also don't know if the "power" notation is the general math notation for constructing tuples from set items, but this is the notation we use at university in computer-science-related courses. I figured this question is more on-topic here than at SO though.)

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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
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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