# Notation: What is the meaning of “$(a,b)$”

I am reading an article about counting hexagonal p-minos (the article is in a combinatorics book) and I saw a notation I don't understand:

$0>(a,b)>-p$ .

$a,b,p$ are integers and so "$>$" means 'bigger', but what can be the meaning of "$(a,b)$" ?

-
perhaps ,"both $a$ and $b$ are less than $0$ and greater than $-p$" . –  pedja Apr 10 '12 at 13:24
agree with pedja. $0>a>-p$ and $0>b>-p$. Brackets to denote difference to $0>a$ and $b>-p$. –  example Apr 10 '12 at 13:25
Does it make sense for $(a,b)$ to mean $\gcd(a,b)$? –  Thomas Apr 10 '12 at 13:29
Can you provide us with some of the (relevant) sentences before and after this? –  Doug Spoonwood Apr 10 '12 at 13:42
I knew from your earlier question. On further examination of Fig. 9, my best guess is that it’s a typo: $(a,b)$ should be $(a',b')$, and the intent is to say that $0\ge a',b'\ge -p$. I’ve not waded through the information on his coordinate system, though, so I make no guarantees! –  Brian M. Scott Apr 10 '12 at 22:43

From the comments it seems pretty clear that it means $0>a>-p$ and $0>b>-p$ (and also that $a$ and $b$ should be $a'$ and $b'$ respectively). The notation is awful, but I think I know why the author doesn't write $0>a,b>-p$: that can be easily misread as $0>a$, $b>-p$, which is a much weaker condition (look closely, if like me you don't see any difference, right-click on the formulas to see the TeX source). I regularly have difficulty avoiding this kind of ambiguity when writing; one could promise to the reader to never to write two conditions separated by just a comma that means "and", but that is an annoying constraint as well, in situations where one needs a somewhat complicated set like $\{(x,y)\in\mathbb R^2\mid x\geq 1, 0\leq y\leq x^2\}$ (not all readers are used to "$\land$" meaning "and"; by the way the perversion of writing "$(a,b)$" instead of "$\gcd(a,b)$" also sometimes takes the alternative form of writing it "$a\land b$"; ah, the delights of laziness…).
It is true that with his private notation the author has managed to make clear that he does not mean $0>a$ and $b>-p$, but at the price of totally obfuscating what he does mean, and all that to save a few keystrokes.
I disagree that never using "," to mean "and" is "an annoying constraint"; I consider an ironclad rule. I would write your example as $\{(x,y)\in\mathbb{R^2} \mid x\ge 1 \text{ and } 0\le y\le x^2\}$. Yes, with "and" spelled out. –  JeffE Apr 11 '12 at 7:51