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$\begingroup$

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: The formula

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`.)2012-07-03
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    Yes! But without the square brackets (not sure why I added those).2012-07-03
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    Though it's kind of bolder and taller but that could just be the book.2012-07-03
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    Ah, I still don't understand what this set is.2012-07-03
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    Wait.. what would be the $i$th complement of a set then?!2012-07-03
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    It might be easier for us, if you could add some information about the book you are reading.2012-07-03
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    It's Proposition 4.2.2 in Stanley's "Enumerative Combinatorics", I think this is a link to the page http://books.google.co.uk/books?id=EvJg1VjIGyMC&pg=PA204&source=gbs_toc_r&cad=4#v=onepage&q&f=false2012-07-03
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    There's a page on notation at the beginning of that google books preview, but it's not mentioned there. Is there a page on notation that it misses out?2012-07-03
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    The notation also appears on page 6.2012-07-03
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    @Ben Thanks, I did that.2012-07-03
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    On page 6 it seems that $\complement_i L(x)$ denotes the $i$-th coefficient of the formal power series $L(x)$.2012-07-03
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    (I deleted a comment earlier which referred to a now-deceased link)2012-09-27

1 Answers 1

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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.2012-07-03
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    What about the notation $[z^n]f(z)$ to denote the coefficient?2012-07-03
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    @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.2012-07-03
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    @PatrickDaSilva It's not *mine*. I saw it in Don Knuth's books, for example, *The Art of Computer Programming*, or *Concrete Mathematics*.2012-07-03
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    @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.2012-07-03
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    They don't. In fact the second edition of Stanley (http://www-math.mit.edu/~rstan/ec/ec1/) uses the $[z^n] f(z)$ notation (see p. 11).2016-02-08