Talk:Catalan number

message: Need to get page number, ISBN for reference to enumerative combinatorics vol 2, and perhaps write up some more examples from there. Dmharvey 19:07, 31 May 2005 (UTC)
thread_title: Untitled

message: *The generating function actually yields the formula for C_n pretty easily, using the binomial formula for the square root term. This should really be described somewhere as the "generating function proof of the formula"; then the "Proof of the formula" section should be relabelled "Bijective proofs of the formula".
*Need to mention the Ballot problem somewhere, which is a generalisation of many of the ideas in this article.
*What was Andre's first name? I can't seem to work this out.
*Need to add this reference:
D. Andre, Note: Calcul des probabilites. Solution directe du probleme resolu par M. Bertrand, Comptes Rendus de l’Academie des Sciences, Paris, vol 105(1887), p. 436., where the reflection principle was first used (I think?)
*I recall reading a long time ago (can't remember where) that the first disucssion of "exceedance" was in a probability setting. Consider a coin tossing game between A and B. They toss a coin 2n times. Heads gives A a point, tails gives B a point. Now, given that they each win n tosses, what is the probability that A is not ahead of B at any point of the game? You could also ask, for any k between 0 and n inclusive, what is the probability that A is ahead of B for precisely k turns? (you need to be a bit careful about how exactly you define when A is ahead.) The point is that it was shown that the answer does not depend on k, and so must be equal to 1/(n+1), which a priori is quite a surprising result. Would be nice to mention this in the History section, but the details need to be found.
*My explanations of the two bijective proofs are a little wordy. Someone please write them better. Thanks.
*Give C_3 examples/pictures for some of the other combinatorial interpretations listed.
*Add more combinatorial interpretations; there are plenty of interesting and accessible ones still left. I don't have a copy of EC vol 2 handy.
Dmharvey 12:07, 1 Jun 2005 (UTC)
thread_title:  To Do list for this article 

message: I have deleted the following from the "history" section.
: It has been shown that the number of possible interpretations of a sentence, function of the number of prepositional groups ("he saw a man on the hill with a telescope"), is the Catalan number series.
It doesn't belong under History. Perhaps it belongs in the list of combinatorial applications. But the text quality needs to be improved, and I don't really know what it's about. Dmharvey Talk 17:11, 6 Jun 2005 (UTC)
replies: text: That's just the number of binary parse trees. Not particularly interesting, seeing as equiparseable sentences with distinct meanings are highly artificial anyway. EdC 10:41, 21 July 2006 (UTC)
thread_title:  "prepositional groups" 

message: Ming An-tu discovered them before but they were published in 1839. 24.203.251.69 03:27, 31 October 2005 (UTC)
thread_title:  History 

message: As an architect I try to understand things with graphics. See:
http://home.versateladsl.be/vt649464/TrapVeelhNummering.PDF
If someone wants to use it I can give a version without text.--Bleuprint (talk) 05:12, 8 February 2008 (UTC)
thread_title:  Graphical relation between Catalan numbers, stairs with slope<45°, triangulated polygons 

message: Dmharvey wrote in the TODO list above:
:Need to mention the Ballot problem somewhere, which is a generalisation of many of the ideas in this article.
I've added Bertrand's ballot theorem to the See also section, of course, some more details are worth including. --Kompik 11:28, 11 March 2006 (UTC)
thread_title: Ballot problem

message: The Catalan Recurrence
C(x)=SIGMA C(i) C(n-1)
has not been solved properly.  A Quadratic eqn. appears out of nowhere.  Can someone explain where it came from?
:It's not "out of nowhere", although the explanation is not very explicit.  It comes from realizing that the sum defines a Cauchy product.  I'll see if I can add something on this. Michael Hardy 20:43, 20 August 2006 (UTC)
replies: text: OK, I've expanded that section, adding a more leisurely derivation of the quadratic equation. Michael Hardy 21:07, 20 August 2006 (UTC)

text: I've added an explicit explanation of that. Michael Hardy 02:07, 22 June 2007 (UTC)
thread_title:  RE: Catalan Recurrence 

message: Perhaps these two problems with Catalan solutions are already in the harder stuff in the article.  If not, do they belong?
1. How many ways can 2n ordered objects be arranged in a 2 x n matrix so that the elements are strictly increasing left to right and top down? (sometimes posed as a photographer arranging 2n people in 2 rows so that the heights increase left to right and front to back)
2. How many ways can n non-intersecting diagonals be drawn in a 2n-gon? (How many ways can 2n people seated at a circular table shake hands with no one's arms crossing)  Jd2718 18:00, 5 November 2006 (UTC)
replies: text: These have been added.--RDBury (talk) 17:53, 8 February 2009 (UTC)
thread_title: Two more Catalan problems

message: : a\,
: a(a+b)=\,
: a(a+b)(a+b+c)=\,
: a(a+b)(a+b+c)(a+b+c+d)=\,
:a\,
:ab+a^2\,
:abc+ab^2+ca^2+2ba^2+a^3\,
:a^2c^2+abcd+abc^2+adb^2+cda^2+ab^3+da^3+2acb^2+2bda^2+4bca^2+3a^2b^2+2ca^3+3ba^3+a^4\,
The sum of the entries of every row is  n! \,
Twentythreethousand (talk) 16:36, 31 May 2008 (UTC)
replies: text: More relevantly, the number of terms is Cn. Similarly, the number of terms in the expansion of (a1+a2+...+ak)n is {n+k-1 \choose n}. Both of these can be proven by mapping the terms to lattice paths.--RDBury (talk) 18:28, 8 February 2009 (UTC)
thread_title:  Catalan numbers 

message: Where should we send the reader who typed "Catalan numbers" into the search box, looking for how to count to ten in Catalan? --Damian Yerrick (talk | stalk) 21:31, 23 August 2008 (UTC)
replies: text: That would probably be Catalan language#Numerals but it doesn't exist yet.--RDBury (talk) 06:25, 4 January 2012 (UTC)

text: This is actually a joke exercise in Stanley's book, asking the student to "Explain the significance of the following sequence: un, dos, tres, quatre, cinc, sis, set, vuit, nou, deu..." Miclugo (talk) 20:21, 23 September 2022 (UTC)
thread_title:  Other kind of Catalan numbers? 

message: The reflection method is described both here and in Bertrand's ballot theorem. I added a more obvious link in the article; should there be an attempt to merge? Also, it's not entirely incorrect to call it André's reflection method, he had the basic idea but without the reflection part. The reflections are essentially only a way of giving a one-one proof that 
{2n\choose n-1} = {2n\choose n+1}.--RDBury (talk) 17:11, 8 February 2009 (UTC)
thread_title:  Ballot problem/second proof 

message: The article says that "A Dyck word is a string .. such that no initial segment of the string has more Y's than X's" then it lists several examples including XXXYYY. But: X, XX, XXX, XXXY are all initial segments of XXXYYY and each of them has more X's than Y's.--Jirka6 (talk) 00:25, 10 April 2009 (UTC)
replies: text: My fault, sorry. Of course it is correct. I switched X and Y.--Jirka6 (talk) 22:18, 13 April 2009 (UTC)
thread_title:  Is the definition of Dyck's word correct? 

message: What do you think?  Is it worthy of inclusion in the article that a Catalan number can be written in terms of a generalized binomial coefficient?:
:\begin{align}C_n
&= \frac{2n!}{n! (n+1)!}
= \frac{(1 \cdot 3 \cdot \ldots \cdot (2n-1))(2 \cdot 4 \cdot \ldots \cdot 2n)}{n! (n+1)!}
= 2^{2n} \frac{(\frac 12 \cdot \frac 32 \cdot \ldots \cdot \frac{2n-1}2)(\frac 22 \cdot \frac 42 \cdot \ldots \cdot \frac{2n}2)}{n! (n+1)!} \\
&= 2^{2n} \frac{\left(\frac{\Gamma(n+1/2)}{\Gamma(1/2)}\right)(n!)}{n! (n+1)!}
= 2^{2n} \frac{\left(\frac{\Gamma(n+1/2)}{(-1/2)\Gamma(-1/2)}\right)}{(n+1)!}
= -2^{2n+1} \frac{\left(\frac{\Gamma(n+1/2)}{\Gamma(-1/2)}\right)}{(n+1)!}
= -2^{2n+1} \binom{n-\frac 12}{-\frac 32}
\,,\end{align}
using the Gamma function.
Thanks -- Quantling (talk) 17:01, 21 April 2010 (UTC)
thread_title:  As a binomial coefficent 

message: I have added a new bijective proof aimed to directly derive the fascinating \tfrac{1}{n+1} factor in the formula. I think it makes a nice complement to the existing bijective proofs as it is not based on a geometric interpretation but instead uses Dyke words and calculates the factor algebraically by comparing coefficients.
I tried several approaches and found this to be the most economical. I made this proof up myself as I could not find a similar one with the same features on the web. Still it should be simple and straightforward enough to be immediately obvious and verifiable by anyone with basic math skills in order to justify its inclusion into wikipedia. If it has been done before (which seems likely), please feel free to add a reference.
Bernhard Oemer (talk) 16:02, 10 January 2011 (UTC)
thread_title:  New non-geometric bijective proof 

message: What the heck is this section refering to?  Where does the name "quadruple factorial" come from?  Is it notable at all?  Right now I'm inclined to delete it because I can't figure out what it's about, but if someone can tell me what it means, then I'd be happy to reverse my position.  (Maybe it would be better as a subsection somewhere?)  --JBL (talk) 13:32, 17 September 2012 (UTC)
replies: text: Hearing nothing after one year, I've removed mentions of the "quadruple factorial". --JBL (talk) 03:10, 16 September 2013 (UTC)
thread_title:  Section "quadruple factorial" 

message: I have just finished some copyediting of the second proof to improve the English and flow of the proof. I tried to preserve as much of the section as I could, but it did require some significant changes to increase clarity and deal with some of the harder parts of the proof. I did not, at this time, put back the historical sentence concerning Andre's reflection method, since there is an editor who objects to that, and so, it should be discussed here. My feeling is that it should be returned since it has encyclopedic value and we are writing an encyclopedia, not a textbook. The statement is of interest and I should point out a comment concerning it made above on this talk page. Bill Cherowitzo  (talk) 06:41, 8 January 2015 (UTC)
replies: text: I would accept one sentence about the author if it provided some new information in addition to the reflection article. The point to discuss the authorship of reflection method, whenever you apply it is stupid verbosity and as any verbosity it was confusing: I could not even understand who is author of what. `Encyclopedia` does not mean that you copy-paste referred material under every reference. Which encyclopedic standard tells you that you must copy only the historic sections but not the whole referenced articles every time you use/refer them? Finally, sane people can distinguish Andreas reflection from Schwarz reflection without notice. Only dumb ones will follow the hyperlink and confuse it with Schwarz reflection. For them, you should notice that Andre's reflection principle is different from both electricity, economics, computer manufactoring, psychology and everything we have in the world (there are  20 reflection principles that you had to enumerate in your article at least). It seems that your purpose was to draw the proof into the stupid verbosity. I do not understand which fluence you are talking about. --Javalenok (talk) 13:09, 11 January 2015 (UTC)
thread_title:  Second proof 

message: https://en.wikipedia.org/wiki/Minggatu
 
"Mingantu" felt bogus to me, so I Googled it, and found the more plausible entry, "Minggatu," above.  My feeling is that this "Mingantu" thingie, and the related renaming of his hometown as "Ming Antu," apparently, by some Chinese bureaucrat, are the work of perhaps well-intentioned but semi-literate functionaries.  The Han government in Beijing is faced with all sorts of ethnic frictions, and they're scurrying around trying to solve them, but clearly some of the people involved are acting without really understanding what they're doing.  (We have the same sort of thing in the West, with all the halfwits who pronounce the Chinese capital a Frenchified "bay-zhhing."  It's furrin, but if I make it sound French that will show people how cosmopolitan and sophisticated I am, right? Feh!)
 
Imho, we ought to go with the way the guy pronounced it himself, and in Roman letters that would be "Minggatu," to the minimum possible distortion.
 
Fixing this will involve inserting the Wikipedia reference as #14, and renumbering the higher footnote numbers, which I'm afraid is beyond my Wiki skills.  Could somebody else maybe do that?
 
David Lloyd-Jones (talk) 12:31, 6 November 2017 (UTC)
replies: text: Um, your feelings about this are irrelevant.  None of the listed sources use this spelling, and if anything, the biography article should be moved to reflect the spellings we have in sources.  --Deacon Vorbis (talk) 13:10, 6 November 2017 (UTC)
thread_title:  Name: Mingantu, Minggatu? 

message: Should there be a mention of inversion of power series? One way to define the Catalan numbers is to say that they are the coefficients in the inverse series of x(1-x), regarded as a power series in x. In other words, if you set y=x(1-x) and solve for x as a series in y, you get them as coefficients. Maybe this follows pretty readily from one of the other descriptions given here. I don't feel confident enough to wade in and make a change to the article.  Ishboyfay (talk) 23:22, 16 November 2017 (UTC)
replies: text: , there seems to already be a blurb about it at Lagrange inversion theorem#Binary trees.  You could always just write a sentence or two that mentions something about it and link to there.  --Deacon Vorbis (talk) 00:25, 17 November 2017 (UTC)

text: On further thought, it's so similar to what appears in Proof 1 that it's not a useful addition. Ishboyfay (talk) 03:50, 18 November 2017 (UTC)
thread_title:  Should there be a mention of inversion of power series? 

message: In the list of applications, applications 2, 3, and 13 are stating the same thing. Also, applications 4 and 12 are redundant. I am going to remove applications 3, 12, and 13 upon consensus.
Plaba123 (talk) 16:23, 15 October 2018 (UTC)
replies: text: First of all, a lot of these are going to be fairly similar, and I think a little redundancy for listing what's counted is actually okay.  That being said, I think 12 (rooted binary trees) is definitely just a repeat of 4 (same thing), so it's safe to remove.  I think #13 (mountain ranges) is probably okay to remove too, but more so because it's really the same as #6 (lattice paths), although this one isn't quite as clear-cut.  But 2 and 3 are probably different enough that they should have their own entries.  If anything, #2 is too similar to #1, but the connection to #3 isn't quite so immediately obvious.  –Deacon Vorbis (carbon • videos) 17:22, 15 October 2018 (UTC)
replies: text:  Thanks for your input. I'll start off by removing #12 (rooted binary trees). Plaba123 (talk) 01:25, 16 October 2018 (UTC)
thread_title:  Redundancy in Applications in combinatorics 

message: The second recursive formula was shifted one to the right,
it returned 1 for C_0, 2 for C_1, 5 for C_2 and so on.
OEIS gives the same recursive definition (https://oeis.org/A000108):
2*(2*n-1)*a(n-1) = (n+1)*a(n)
I see that my change was reverted, could it be incorporated? Pranomostro (talk) 21:06, 18 February 2019 (UTC)
replies: text: Side note first: there's certainly nothing wrong with notifying someone on their talk page of a discussion on an article's talk page, but if you want to get someone's attention on an article talk page post directly, you can just WP:PING them (I'll generally use either the  or  template, depending).  Anyway, about the recurrence relation, you might want to double check that you're using the right value for .  The formula that's already here is correct and agrees with what's listed at the OEIS.  –Deacon Vorbis (carbon • videos) 02:11, 19 February 2019 (UTC)

text: You are mistaken, and the version in the article is correct.  Your error comes from replacing n+1 and n with n and n-1 in the subscripts, but not also making the same substitution for the other copies of n.  The simplest way to verify that you are mistaken is to plug in n=0 in the version in the article.—JBL (talk) 23:42, 19 February 2019 (UTC)
thread_title:  Second recursive formula 

message: Using WA, one can use the following query:  (5*4^(n-3) * (Pochhammer 7/2,n-3)) / (Pochhammer 5,n-3) for n=3,10
:: alternately,  Table[ CatalanNumber@ n, {n, 0, 100}]    --Billymac00 (talk) 18:44, 3 September 2020 (UTC)
replies: text:  And? --JBL (talk) 19:34, 3 September 2020 (UTC)
thread_title:  via Wolfram Alpha rising factorial 

message: Thank you @JayBeeEll for your recent concern about my edit to this article.  Would you please give more information about your comment "this is not an example of what is being discussed"?  The paragraph is about the number of associative ways to multiply an ordered set of n+1 objects.  For the matrix chain multiplication problem, the n+1 objects are matrices.  To me, it is the premier example where people care about the association order for the multiplications.  Thank you —Quantling (talk | contribs) 17:38, 8 February 2021 (UTC)
replies: text: The paragraph is about the number of ways to associate, whereas mcm is the context in which people care about different associations.  The latter is not an example of the former.  In fact, the connection between the Catalan numbers (the topic of this article, and the common thread that binds all the different examples in that section together) and the mcm problem is rather weak: one does not need to know a priori how many parenthesizations there are in order to solve mcm, and there's nothing about mcm that is more Catalan-y than the generic observation about possible associations of any binary operation.  I did not remove the link from the article -- it's now in the see also section -- but if you really think it would be better in the bullet point, we could talk about other phrasing.  One possibility might be to extend the parenthetical in the previous sentence: "(or the number of ways of associating n applications of a binary operator, as in the matrix chain multiplication problem)". --JBL (talk) 17:54, 8 February 2021 (UTC)
thread_title:  Relation to matrix chain multiplication 

message: If any talk-page watcher has any interest in taking this to a GA or improve the overall quality, I will appreciate their collaboration. Will start drafting at User:TrangaBellam/Catalan Number. Thanks, TrangaBellam (talk) 18:00, 21 April 2022 (UTC)
replies: text: It needs a lot of work, but I am definitely interested.  End-of-semester busyness may be an issue for me for the next few weeks, though. --JBL (talk) 18:02, 21 April 2022 (UTC)
thread_title:  Taking this to GA and maybe, beyond 

message: I think using "we" in the proofs is kinda weird and should be removed  — Preceding unsigned comment added by 2604:3D09:1580:9600:C884:4058:E2C4:3F13 (talk) 03:19, 17 May 2022 (UTC)
thread_title:  We 

message: @Darcourse et al.  There is use of the method of images in the second proof of the Catalan numbers formula.  While the method of images is used for continuous cases, such as energy potentials in the presence of boundary conditions, it is also used for discrete cases.  A random walk that starts at the origin on the real line, that takes steps of ±1 , and that is prohibited from going to −1 — in our case the number of '(' minus the number of ')' cannot be −1 — can be modeled by adding the random walk to a "negative" of the random walk that starts at −2.  For any given number of steps, both the random walk from the origin and the one from −2 have equal chance of ending up at −1, and thus adding that "negative" "image" that starts at −2 to the original random walk from the origin ends up zeroing all values at −1, which is exactly what we need when we are enumerating Catalan numbers.
I propose to mention this in the section on the second proof, via something like
:
but I am open to alternatives.  Thanks —Quantling (talk | contribs) 14:57, 16 October 2023 (UTC)
replies: text: The MoI article is way too technical for the level the Catalan number article is aimed at, and the connection is no way obvious. You could add it to 'see also', but I can't see how it is related at all. Darcourse (talk) 03:55, 17 October 2023 (UTC)
replies: text: The Method of images article is lacking an example that is a discrete case.  I plan to make edits there and then revisit the present discussion. —Quantling (talk | contribs) 12:58, 17 October 2023 (UTC)
thread_title:  Method of images