Question on meaning of a symbol: long thin C

I don't know how to write it in $\LaTeX.$ It is a tall skinny bold C. This is the context: A set is defined by:

where $\complement\atop{\smash \scriptstyle i}$ is the thing I don't understand. The $i$ is actually directly underneath the weird $C$ in this case. Can anyone explain what this means?

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Does it look like $$E_f = \{i:{\complement_i} L(x)\ne 0\}$$? (The $\complement$ is \complement.) – MJD Jul 3 '12 at 14:41
Yes! But without the square brackets (not sure why I added those). – 098765 Jul 3 '12 at 14:43
Though it's kind of bolder and taller but that could just be the book. – 098765 Jul 3 '12 at 14:44
Wait.. what would be the $i$th complement of a set then?! – user2468 Jul 3 '12 at 14:45
On page 6 it seems that $\complement_i L(x)$ denotes the $i$-th coefficient of the formal power series $L(x)$. – martini Jul 3 '12 at 15:06

1 Answer

It's the coefficient operator. It extracts the ith coefficient of the Taylor expansion. This is used a lot in combinatorics with generating functions.

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Thanks for this. I've just seen it explained on page 3 as well. – 098765 Jul 3 '12 at 15:07
What about the notation $[z^n]f(z)$ to denote the coefficient? – Frank Science Jul 3 '12 at 15:14
@Frank Science : It may happen that a standard notation is not the best one, but the main property that's interesting of a standard notation is that it is 'standard', i.e. everyone uses it and everyone is used to it. Sometimes this is because everyone tends to like it after working for a while in the field where the notation is used. It doesn't mean your idea is good or wrong, just that if you want to make a new notation popular, usually you prove a few things in that field of research that makes the use of this new notation more useful than the previous one. – Patrick Da Silva Jul 3 '12 at 15:18
@PatrickDaSilva It's not mine. I saw it in Don Knuth's books, for example, The Art of Computer Programming, or Concrete Mathematics. – Frank Science Jul 3 '12 at 15:19
@Frank Science : I didn't know. But still, my comment applies ; perhaps people working with generating functions prefer the $\complement$ notation to the $[z^n]f(z)$ notation. – Patrick Da Silva Jul 3 '12 at 18:12