Talk:Stirling numbers of the first kind

From Wikipedia, the free encyclopedia

This article incorporates material from PlanetMath, which is licensed under the GFDL.

[edit] Unclear

I removed the following text from the section on generating functions. The reason I removed this is because half of this is as opaque as mud, and the other half doesn't contribute anything additional to what this section already states. linas 01:36, 29 December 2006 (UTC)

The generating functions of the signed and unsigned Stirling numbers may be constructed from combinatorics. The fundamental theorem of combinatorial enumeration applies (labelled case).A permutation is a set of cycles, and hence the set \mathcal{P}\, of permutations is given by
\mathcal{P} = \mathfrak{P}(\mathcal{U} \mathfrak{C}(\mathcal{Z})),
where the singletion \mathcal{U} marks cycles. This decomposition is examined in some detail on the page on the statistics of random permutations.

What the heck do those mathfrak blackpage symbols mean? The page on "statistics of random permutations" does not explain these symbols, and even if it did, I'm not sure their use here would be appropriate.

Translating to generating functions we obtain the mixed generating function of the unsigned Stirling numbers of the first kind:
G(z, u) = \exp \left( u \log \frac{1}{1-z} \right) = \left(\frac{1}{1-z} \right)^u = \sum_{n=0}^\infty \sum_{k=0}^n \left|\left[\begin{matrix} n \\ k \end{matrix}\right]\right| u^k \, \frac{z^n}{n!}.
Now the signed Stirling numbers of the first kind are obtained from the unsigned ones through the relation
\left[\begin{matrix} n \\ k \end{matrix}\right] = (-1)^{n-k} \left|\left[\begin{matrix} n \\ k \end{matrix}\right]\right|.
Hence the generating function H(z,u) of these numbers is
H(z, u) = G(-z, -u) = \left(\frac{1}{1+z} \right)^{-u} = (1+z)^u = \sum_{n=0}^\infty \sum_{k=0}^n \left[\begin{matrix} n \\ k \end{matrix}\right] u^k \, \frac{z^n}{n!}.

I cut this too, as it appears to be a trivial manipulation of the content that is already there, and doesn't add any useful information. linas 01:36, 29 December 2006 (UTC)

[edit] More removed text

I'm removing more text:

This formula holds because the exponential generating function of the sum is
H(z, -1) = \frac{1}{1+z} \quad \mbox{and hence} \quad n! [z^n] H(z, -1) = (-1)^n n!.

First of all, I have no clue what [zn] is supposed to be or mean, so this proof appears to be utterly bogus. Second of all, proofs like this don't belong in article space, they should really be moved to a proof subpage; see Category:Article proofs for examples. linas 01:45, 29 December 2006 (UTC)

Also, for same reasons:

This relation holds because
[u^k] H(z, u) = [u^k] \exp \left( u \log (1+z) \right) = \frac {\left(\log (1+z)\right)^k}{k!}.

[edit] Generating function

I know what it means. It's a common way to denote generating functions. When writing out H(z,u) as an expansion in z, then [z^n]H(z,-1) is the coeffient of the term with z^n with fixed u=-1. Likewise, [u^k]H(z,u) is the coefficient of u^k when expanding H(z,u) in u. Imho, removing something when you DON'T understand it is not correct (at most arrogant :p). It's better to only remove things when you are an expert on the subject. Carlo Wood 06:13, 14 February 2007 (UTC)

Yes, well, the conversation might be more productive if the actual issues were addressed; being rude and insulting is not helpful. I'm not asking for much, I'm just asking for a definition of the terminology; this is common and accepted practice not just in WP but all academic writing, and is certainly spelled out in WP style guides. Not much point to an article if only the experts understand what it says. linas 04:28, 19 March 2007 (UTC)
Dear Linas, I am definitely not angry, as you say on my talk page, since you are working very hard to make Wikipedia the best it can be. As for the article on Stirling numbers and EGFs, this is what I wrote in December 2006: "... the page also includes a link to Analytic combinatorics - Symbolic combinatorics., written by two of the most brilliant people in the field. May I respectfully suggest you have a look at it. You might find it exciting reading. As for the Stirling number/Generating function stuff, I agree it should not go into the main article, but we do not want to miss out on the powerful combinatorics techniques formalized in the eighties." So you see, this has nothing to do with me, or with proving anything. I simply believe that these developments from the eighties, which most specialists would agree count among the most significant in the field, should definitely find a place in Wikipedia. This "claim" is easy to verify. Just ask anyone in the math dept. of a university near you. Now, on the writing itself, which you say is flawed. Well, I couldn't agree more, as far as my contribution is concerned. I am not good at expository writing. But that's what the Wikipedia edit model is for, right? I hope you or someone else who is interested and qualified will make the necessary corrections and turn the article into a resource that is useful for everyone. (BTW the coefficient extraction operator has been documented on the formal power series page for some time now.) -Zahlentheorie 13:56, 19 March 2007 (UTC)