Talk:Eulerian polynomials

Name of the triangle

Daniel Forgues wrote:

The triangle of Eulerian numbers (preferred term) is also called Euler's triangle (this term has too much similarity with Euler's triangle theorem or Euler's triangle formula, which are unrelated).

I think this wording is not appropriate. It is too subjective and potentially controversial. I think it is not the goal of this wiki to give mathematicians instructions how to name their objects of interest in a better way. Clearly you can discuss this and make such proposals. However, I think such a discussion takes a better place on your personal blog. The term 'Euler's triangle' is well established and it is used for example in the highly acclaimed and standard-setting Concrete Mathematics by Graham, Knuth and Patashnik (Table 254). Therefore I will undo the last modification.

Peter Luschny 17:57, 3 August 2010 (UTC)

You are right about this!

In Eulerian numbers, triangle of, I also replaced:

The triangle of Eulerian numbers (preferred term) is also called Euler's triangle (this term has too much similarity with Euler's triangle theorem or Euler's triangle formula, which are unrelated).

by:

The triangle of Eulerian numbers is also called Euler's triangle.

Thanks

— Daniel Forgues 18:18, 3 August 2010 (UTC)

I think mentioning and/or linking to these unrelated notions (Euler's triangle theorem or Euler's triangle formula) would not be a bad thing.— M. F. Hasler 19:38, 3 November 2011 (UTC)

Citing W|A

Hi Daniel!

Thank you for the traditional exponential generating function for the Eulerian polynomials, which was given explicitly by Euler himself in his memoir.

This egf is, of course, also stated in the references given (see Faota, (3.1)). And, of course, also on OEIS in the form suitable for the Eulerian numbers.

Therefore I see no reason to cite Wolfram Alpha here. (Alpha is in my opinion a reference which should be given sparingly, only if no other or better alternative is available.) I will delete the link to the Alpha as being irrelevant, but of course retain Euler's egf. I hoped to find out who else is reading this page...

Peter Luschny 15:20, 6 August 2010 (UTC)

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I found it nifty to see Wolfram Alpha calculate in real time (it is not just a static curated knowledge base, it also uses Mathematica) the Eulerian polynomials (click for more terms at: Series expansion at x=0:), it is cool to see it in action and it will also make the verification process (by the official reviewers) faster, I think.

Thanks

— Daniel Forgues 16:14, 6 August 2010 (UTC)

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I see.

If I like to tinker with some formula I find on a web page I copy it to my Maple. It would never occur to me to feed it to Alpha. The possibilities of interaction are severely limited.

Moreover, I think 99% of the persons which are interested in esoteric stuff like 'Eulerian polynomials' have a CAS and would proceed similarly. In the case you have no CAS I recommend to get Sage, a very fine free open-source mathematics software system. http://www.sagemath.org/

In this situation I see no reason to clutter up the reference section with links to a business company -- at least not as long as they do not pay me a promotion fee. In fact I even like to avoid the appearance that they do.

Peter Luschny 22:04, 6 August 2010 (UTC)

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I have put a lot of links to Wolfram Alpha throughout my figurate numbers pages since it provides a very quick way to verify formulas (especially handy to see the sequence terms generated by a generating function or a formula) for errors or typos at any later time via a simple click (especially, I think it could save time for the editorial board when they go through the process of reviewing the wiki pages for approval). I guess this is free promotion for a private company, but then we get to access their computing resources to quickly verify formulas for free... I will ask Neil Sloane and David Applegate what they think about it (should I stop this practice, or just use it more sparingly, and whether or not I should remove all the numerous links that I already put throughout?)

Thanks

— Daniel Forgues 00:49, 7 August 2010 (UTC)

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I just sent this email to Neil Sloane and David Applegate to get feedback about the usage of Wolfram Alpha:

Is it a good thing or not to use a lot of links to Wolfram Alpha for quick verification of formulas at any time now or later on for pages in the Wiki

Daniel Forgues to David, Sloane

show details 9:08 PM (8 minutes ago)

I have put a lot of links to Wolfram Alpha throughout my figurate numbers pages since it provides a very quick way to verify formulas (especially handy to see the sequence terms generated by a generating function or a formula) for errors or typos at any later time via a simple click (especially, I think it could save time for the editorial board when they go through the process of reviewing the wiki pages for approval.) I guess this is free promotion for a private company, but then we get to access their computing resources to quickly verify formulas for free... I will ask Neil Sloane and David Applegate what they think about it (should I stop this practice, or just use it more sparingly, and whether or not I should remove all the numerous links that I already put throughout?)

For example:

[http://www.wolframalpha.com/input/?i=%28t-1%29%2F%28t-\exp%28%28t-1%29x%29%29 Generating function of Eulerian polynomials], i.e. (t-1)/(t-\exp((t-1)x)), then at (Series expansion at x=0:) click for more terms to quickly verify that the generating function is correct (no error or typo) and that you effectively get the Eulerian polynomials.

Cf. [1].

Thanks

— Daniel Forgues 01:27, 7 August 2010 (UTC)

Literature reference

Richard Mathar believes this may be the first reference to the OEIS Wiki in the standard literature:

E-U. Gekeler, "On the zeros of Goss polynomials." Trans. Am. Math. Soc. 365 93012) 1669-1685.

Alonso del Arte 04:29, 19 April 2013 (UTC)

Using the {{cite paper}} template:

Gekeler, E-U. (2012). “On the zeros of Goss polynomials”. Trans. Amer. Math. Soc. 365 (3): pp. 1669–1685. doi:10.1090/S0002-9947-2012-05699-6. http://www.ams.org/journals/tran/2013-365-03/S0002-9947-2012-05699-6/home.html. 

— Daniel Forgues 02:18, 20 April 2013 (UTC)

Style

The current page style, which has a textbook style and is obviously intended for book printing, definitively looks great (although it leaves a lot of unused blank space to the right of a computer's screen window), is different from the rest of OEIS Wiki, which is to use the window's width and automatic flow of text. I would prefer to have the same style as the rest of the wiki, for consistency. — Daniel Forgues 23:53, 21 April 2013 (UTC)

I would greatly prefer a style which does not have a fixed width. Let readers decide what width they want (by resizing their browser window)! Charles R Greathouse IV 02:28, 22 April 2013 (UTC)

Now, it (the draft copy, that is) has the same style as the rest of the wiki. — Daniel Forgues 02:59, 22 April 2013 (UTC)

I feared from the beginning that sooner or later the layout with fixed width will be killed, by whatever arguments. So I was pleasantly touched when Neil last year wrote that he had visited the "very nice pages from your blog" (which all have this style). Therefore just some remarks how I see this matter of style:

It's about readability, and there are well established results in the psychology of perception and typography which give clear guidelines in this matter; and they apply to books as well as to online media.

First note that fixed text width is the standard today for Html-based presentations. This has definitely changed from the dark age layouts in the 1990.

Convince yourself by looking at some mainstream media like

• The New York Times knows this and uses fixed width NYT
• Le Figaro knows this and uses fixed width Figaro
• Social media know this and use fixed width MathOverflow
• Educational media know this and use fixed width AMSFeature
• Bloggers know this and use fixed width johncarlosbaez
• Encyclopedias know this and use fixed width britannica
• Even fingerprint archives know this and use fixed width OEIS

The only major software out there which does not use fixed width as default are the wikis, a software based in the internet world of 1995. But it can be easily adapted by content writers by using Html-tags like <div>. So this is no real problem. Why people insist on the strange charm of 1990's Html is beyond my understanding.

It is not about 'the freedom of the user to choose'. The user still can zoom in or out into the page. It's about a good reading experience.

And No, you cannot achieve the same reading experience by resizing the browser window. Look for instance at this screenshot of the current draft. A senseless identity. The browser window is rightly adjusted for good reading and - boom - you cannot read the last formula without scrolling left-right. And this is just one immediate example.

Yes, fixed width will demand more attention from the writer. They will have to look for ways to fit into the given frame. And that is a good thing. Man who care, like Don Knuth, sometimes even reformulate their sentences to make them fit into a line or a paragraph.

Senseless monstrosities which do not fit on a book page should not be produced just because obsolete software allows the plagued user to be forced to scroll his browser window horizontally or give his browser window more room on his desktop than he wishes.

But I will not insist on my point of view and so I will drop fixing the text width knowing to do a disservice to the readers of this page. Besides readers are also invited to the version on my homepage.

Daniel's comment "Now, it has the same style as the rest of the wiki." made me look up one of my favorite YouTube videos.

Peter Luschny 17:49, 23 April 2013 (UTC)

Consistency (or conformity as shown in the video) is not a good thing, I agree. So I'll revert to your style, which actually looks better. — Daniel Forgues 08:29, 24 April 2013 (UTC)

We could ask Neil Sloane to modify the style sheet for the wiki to wrap all pages with, e.g.

<div style="width: 580px; text-align: justify; margin-left: 75px; margin-top: 50px; margin-bottom: 50px;"> ... </div>

— Daniel Forgues 08:40, 24 April 2013 (UTC)

The wiki would look much nicer and professional with your fixed-width justified style, I totally agree. It would be great to have this as a global style sheet.

<div align="center"> <div style="width: 580px; text-align: justify; padding-left: 75px; padding-right: 75px; padding-top: 50px; padding-bottom: 50px; border: 1px solid lightgray;"> ... </div></div>

— Daniel Forgues 08:59, 24 April 2013 (UTC)

I tried the fixed-width justified style on Fermat's last theorem. — Daniel Forgues 09:50, 24 April 2013 (UTC)

The style is now

<div align="center" style="background: #f9f9f9; padding: 25px;"> <div style="width: 580px; text-align: justify; padding-left: 75px; padding-right: 75px; padding-top: 50px; padding-bottom: 50px; border: 1px solid lightgray; background: white;"> ... </div></div>

— Daniel Forgues 11:51, 24 April 2013 (UTC)

I created the {{OEIS Wiki style}} template. I defined the textbook style (a slightly edited version of your original style) in the template and applied it to this talk page (although it is likely not appropriate for talk pages). So now it is easy to keep track of styled pages and to revert or modify the style. — Daniel Forgues 02:24, 26 April 2013 (UTC)

See https://oeis.org/w/index.php?title=Talk:Eulerian_polynomials&oldid=1576949 for a previous version of this talk page styled with the terminal style! :-) — Daniel Forgues 03:09, 27 April 2013 (UTC)

Display of continued fraction

Daniel, you made the proposal to write:

${\begin{aligned}\left(1-t+{\underset {k=1}{\overset {\infty }{\rm {K}}}}~{\frac {k^{2}\,x\,t^{2}}{(k+1+kx)\,t-1}}\right)^{-1}&={\cfrac {1}{1-t-{\cfrac {x\,t^{2}}{1-(2+x)\,t-{\cfrac {4x\,t^{2}}{1-(3+2x)\,t-{\cfrac {9x\,t^{2}}{1-(4+3x)\,t-{\cfrac {16x\,t^{2}}{\ddots }}}}}}}}}}\\\end{aligned}},$

where $\scriptstyle {\underset {k=1}{\overset {\infty }{\rm {K}}}}{\frac {b_{k}}{a_{k}}}$ is Gauss' Kettenbruch notation for continued fractions.

Well Daniel, I hope you agree that there is no mathematical content in this pseudo-equation, therefore I will drop it. It is just notational play which distracts from the content. But perhaps you can include it in your "table of nice continued fractions". Now which one of the two forms to choose? "Gauss' Kettenbruch notation" is not widely used by English typesetters as you rightly wrote at another place (and also not in other countries) therefore I think it is better to stick to the staircase notation; perhaps also because it better reflects Euler's use of continued fractions. Peter Luschny 17:55, 23 April 2013 (UTC)

The notation having proposed by Gauss, I wonder why it didn't stick. — Daniel Forgues 08:14, 24 April 2013 (UTC)

With fixed-width, long formulas really spills out on the right like you say. If we adopt fixed-width justified style, lots of formulas will have to be broken into many lines. Also, lots of tables will be much too wide (not a problem then, we just make an exception for the width of those pages, we don't need absolute consistency throughout...) — Daniel Forgues 12:00, 24 April 2013 (UTC)

For Eulerian numbers, triangle of, the triangle and the tables are too wide, so the page would have to be kept wide. — Daniel Forgues 12:39, 24 April 2013 (UTC)