Talk:Stirling numbers of the first kind

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)

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 $$[z^n]$$ 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!}.$$


 * The symbol [u^k] means: extract the coefficient of u^k. However, I do agree the article is awful. Zaslav (talk) 22:14, 25 December 2011 (UTC)

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)


 * As a professional combinatorialist who knows something about the subject, I believe most experts don't agree that these specialized developments "count among the most significant in the field", although they would not discount them either. There is an article on Symbolic combinatorics according to Flajolet-Sedgewick, and their highly technical, not widely known, and usually unnecessary notation should be confined to that article.  Zaslav (talk) 20:08, 18 November 2012 (UTC)

Inaccurate!!
Using the formulae provided, I was unable to reproduce the table provided on the page itself. It later turned out that in several places negative signs were missing ... For example
 * $$\left[{n+1\atop k}\right] = n \left[{n\atop k}\right] + \left[{n\atop k-1}\right]$$

should be


 * $$\left[{n+1\atop k}\right] = \left[{n\atop k-1}\right] - n \left[{n\atop k}\right]$$

Also


 * $$\left[{n\atop n-1}\right] = {n \choose 2} $$

should be


 * $$\left[{n\atop n-1}\right] = -1 {n \choose 2},$$

And
 * $$\left[{n\atop n-3}\right] = {n \choose 2} {n \choose 4}.$$

should be


 * $$\left[{n\atop n-3}\right] = -1 {n \choose 2} {n \choose 4}.$$ —Preceding unsigned comment added by 217.34.163.161 (talk) 10:57, 22 July 2008 (UTC)


 * The formulae in the article are correct, since $$\left[{n\atop k}\right]$$ is the unsigned Stirling number. /Pontus (talk) 10:44, 22 August 2008 (UTC)

Explicit formula
This section is quite unclear, badly formated, and not sourced. Are you sure it is needed ? Compare with the French version fr: Nombre_de_Stirling for more interesting formulas... --Dfeldmann (talk) 06:15, 2 March 2010 (UTC)
 * Because the formula is more complicated than standard one and seems not to be reviewed, I don't think it is important. See also Talk:Stirling number and Talk:Stirling number. The IP user added this kind of formula to French Wikipedia, Japanese Wikipedia (my home wiki), etc. Some of these are already removed. --白駒 (talk) 22:02, 18 March 2010 (UTC)

I checked the source cited, and the formula contained in the book looks nothing like the formula on the page as it shows up on edit preview. I also think it's poorly formatted because it shows up as a TeX parse error in the page itself. 208.95.49.166 (talk) 23:00, 8 February 2014 (UTC)

Notation for Stirling numbers
I believe (from experience) that the most common notation for Stirling numbers of the first kind nowadays is s(n,k). The notation with brackets seems to be relatively rare. It also conflicts with the notation for Gaussian coefficients. Hence, I suggest changing the notation in this article. Is there a reason not to do so?

It is also a good idea to get rid of the "symbolic combinatorics" notations that contribute mostly confusion. The article on "symbolic combinatorics", which is not a widespread method but only one theory of combinatorial enumeration, is the right place for these notations. Zaslav (talk) 20:15, 18 November 2012 (UTC)


 * Sounds fine to me. Stanley and Bona both use c(n, k) for the unsigned numbers (I guess c for "cycle"?).  By the way, I was looking it over and thinking that the "explicit formula" section seems like a lot of space for a pretty marginal formula; what do you think? --JBL (talk) 20:56, 18 November 2012 (UTC)


 * The "explicit formula" is an important one but the way it was written is uselessly complicated. I've simplified it but I'm not sure it's correct.  I will look into that later, unless someone beats me to it. Zaslav (talk)


 * Bracket notation is used in Concrete Mathematics and gaining popularity since then. Maxal (talk) 04:49, 19 November 2012 (UTC)


 * I don't think bracket notation has become very common. Disadvantages: it is space-consuming, and conflicts with common notation for Gaussian coefficients.  I'm disinclined to keep it.  But let's keep the discussion going for a while.  Zaslav (talk) 03:33, 20 November 2012 (UTC)


 * Since bracket notation has not gained predominance and, worse, is in outright conflict with the standard bracket notation for Gaussian coefficients, and since there has been no further comment objecting, I will change the notation to s(n,k), which I believe to be more prevalent than other notations these days (based on admittedly limited reading), with the comment that there are other notations. If anyone has a comment about this, please write it here, and thanks. Zaslav (talk) 02:51, 15 November 2013 (UTC)

Recurrence relation error?
I think the recurrence relation is


 * $$ \left[{n+1\atop k}\right] = k \left[{n\atop k}\right] + \left[{n\atop k-1}\right]$$

Since you can union the new element with any of the k sets. I made a function with this recurrence and it worked fine. If I'm missing something, please tell me. — Preceding unsigned comment added by Bla Maeda (talk • contribs) 04:34, 7 May 2013 (UTC)
 * Were you thinking of Stirling numbers of the second kind? Xanthoxyl  &lt; 14:10, 7 May 2013 (UTC)
 * I believe what you meant was
 * $$\left[{n+1\atop k}\right]=n \left[{n\atop k}\right]+\left[{n\atop k-1}\right]$$
 * which is the recurrence relation for unsigned Stirling numbers of the first kind. Always multiply the row (n), not the column (k). I hope this clears up the confusion. 208.95.49.166 (talk) 23:14, 8 February 2014 (UTC)

The current example list pyramid of numbers is abs(s(n,k)) not s(n,k), you get wrong answers if you use unsigned values in the recurrence relation from the table, btw, should contain negative values! The recurrence relation seems fine otherwise. — Preceding unsigned comment added by Goofyseeker311 (talk • contribs) 14:02, 24 October 2022 (UTC)