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My question is about this pair of sentences near the top of Richard Stanley's Introduction to Hyperplane Arrangements.

"We also write $[t^{k}]χ(t)$ for the coefficient of $t^{k}$ in the polynomial or power series $χ(t)$. For instance, $[t^{2}](1 + t)^{4} = 6$."

I don't know why they are obtaining polynomials as terms of the power series and I don't know why χ(t) is a coefficient of the polynomial or power series that is χ(t).

sqazcdaaasdfsadfa2
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    You don't know what type of polynomial $(1+t)^4$ is? – Thomas Andrews Dec 04 '22 at 23:33
  • Just how can you write χ(t) for a coefficient of the polynomial that is χ(t)? – sqazcdaaasdfsadfa2 Dec 04 '22 at 23:52
  • You don't write $\chi(t)$ for the coefficient, we write $[t^k]\chi(t).$ The $[t^k]$ indicates which coefficient we are seeking. – Thomas Andrews Dec 05 '22 at 00:15
  • Note, this is purely notation, not. statement of fact. It just means $[x^n]f(x)$ is the coefficient of $x^n$ in the Taylor series for $f.$ So you'll see: $[z^3]\log(1+z)=\frac13,$ or $[w^n]\frac{1}{(1-w)^2}=n+1.$ Sometimes you'll even see slight variants: $[x^n/n!]e^x=1$ or Taylor series around another point $[(x-2)^3]f(x)$ or functions of multiple variables: $[x^n] f(x,y)=g(y).$ – Thomas Andrews Dec 05 '22 at 16:03
  • i thought that it was a term in the polynomial by mistake. it's fine now... – sqazcdaaasdfsadfa2 Dec 21 '22 at 23:53

2 Answers2

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You are probably overthinking this. We have $(1+t)^4 = 1 + 4t + 6t^2 + 4t^3 + t^4$, and the coefficient of $t^2$ is 6. So $[t^2](1+t)^4 = 6$.

Ted
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  • I didn't get it because it looks like a polynomial that's equal to 6... when i tried and figure out what that would be for i decided to post here. Thanks. – sqazcdaaasdfsadfa2 Dec 21 '22 at 23:45
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Perhaps you are confused because Stanley just previously defines $[n]=\{1,2,\dots,n\},$ and this statement about $[t^k]\chi(t)$ is broken out of the list of notations, so it looks like an application of the prior notation.

Stanley here is defining another notation, separate from the previous $[n].$ This notation also uses square brackets, so it might be confusing to see the two defined right next to each other.

From a analytic point of view, we might write:

$$[t^n]f(t)=\frac1{n!}f^{(n)}(0),$$ where $f^{(n)}$ is the $n$th derivative of $f.$

Now, technically, this only applies to analytic functions - functions $f$ whose Taylor series converges to $f$ in some open region around $0.$ But mathematicians also deal with abstract power series, where we don't care about convergence for particular values $x.$

For example, $$f(x)=\sum_{n=0}^{\infty} n^nx^n$$ is a power series that does not converge for any non-zero real $x,$ but it is still an abstract power series, and we still say $[x^k]f(x)=k^k.$

This is most useful if you can write your polynomial or power series in a closed form. For example:

$$[x^k]\frac1{(1-x)^{n+1}}=\binom{n+k}{n}$$

or $$[y^k]e^{ay}=\frac{a^k}{k!}$$

Thomas Andrews
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