Catalan numbers and numbering of stairs
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From: bleuprint on 22 Jul 2010 01:47 I just discovered that this Pascal like triangle in the attachment of the 1st post, is simply called the Catalan Triangle. But I still have the following question: Why is the sum of the numbers above each stair UNIQUE? Are there some references about this property? Patrick Labarque
From: Gerry on 22 Jul 2010 07:24 On Jul 22, 7:47 pm, bleuprint <patrick.labar...(a)base.be> wrote: > I just discovered that this Pascal like triangle in the attachment of the 1st post, is simply called the Catalan Triangle. > But I still have the following question: > Why is the sum of the numbers above each stair UNIQUE? > Are there some references about this property? > Patrick Labarque I don't know what you mean by "the 1st post." The 1st post to sci.math was well over 20 years ago. I don't know what you mean by "attachment". Posts to sci.math don't have attachments. But I can tell you that the sum of the numbers above each stair is unique because it is THE sum of the numbers above whichever stair. Whenever you have a bunch of numbers, the sum of that bunch of numbers is unique. It would be highly inconvenient if one bunch of numbers could have more than one sum. But maybe that's not what you meant. -- GM
From: bleuprint on 22 Jul 2010 04:01 The 1st post in this thread (7/20/10) HAS an attachment (TrapVeelhkNummering.pdf 58K). In the drawings you can see that each integer>0 can be associated with ONE "stair-path"(*), and vice versa. So my question is why, and where can I find more references about this property? Patrick The "stair-paths" go along the grid lines, start at 0,0 and ends at n,n and stay below the diagonal between both points (I don't know the exact name of such paths).
From: Dave L. Renfro on 22 Jul 2010 10:02 bleuprint wrote: http://mathforum.org/kb/message.jspa?messageID=7131726 >> The catalan numbers can be associated with the different >> stair steps with a slope of 45° or less. When drawing such >> stairs on a grid it's possible to make a Pascal like triangle >> on that grid so that the sum of the numbers above the stair >> give an integer in such a way that each number is UNIQUE for >> each stair and so that each integer>0 can be associated with >> ONE stair. (See attachment). >> Has this arrangement a name? Where can I find more about it? >> Bleuprint bleuprint then wrote (in part), in reply to Gerry: > The 1st post in this thread (7/20/10) HAS an attachment > (TrapVeelhkNummering.pdf 58K). In the drawings you can > see that each integer>0 can be associated with ONE > "stair-path"(*), and vice versa. So my question is why, > and where can I find more references about this property? When you post to sci.math via Math Forum and include an attachment, the post (even with the attachment omitted) doesn't seem to go anywhere except to the Math Forum sci.math archive web pages. In particular, it doesn't show up at google's sci.math archive. Regarding your question, I suspect Chapter 9: "Lattice Paths and Catalan Numbers" in the following book will help: Thomas Koshy, "Catalan Numbers with Applications", Oxford University Press, 2008, 440 pages. http://books.google.com/books?id=MqPLSivdBDAC&pg=PA259 http://www.amazon.com/dp/019533454X Also, I believe there was an article about lattice paths and Catalan numbers in one of the 2006-2008 issues of "Pi Mu Epsilon Journal" (I don't remember the title of the paper, however). Dave L. Renfro
From: bleuprint on 22 Jul 2010 11:31
As Dave pointed out that some mirror sites of the Mathforum don't give the attachments (they mirror the ears but not the eyes or our nose)I try to explain it with text. Look to the (small) "Catalan triangle" below (*). . . 5 . 2 5 1 2 3 1 1 1 1 The first number left below is 1. Each other number is equal to the sum of the number below and the number to the left of it. The catalan numbers are formed on the diagonal and the sum of each collumn gives also the catalan numbers. We draw now a stair on that grid from left below to upper right without crossing the diagonal. . . 5| . 2 5| + a horizontal line under 5 1 2|3 + a horizontal line under 2 1 1|1 1 + a horizontal line under the two 1's to the left. The sum of the numbers above (or left of) the stair is equal to 1+1+1+2+2+5+5=17. Let's call it the grid-sum. The property (proof!) I'm looking for is the following: "Each stair gives a different grid-sum, and for each integer there exists ONE stair with that grid-sum." Patrick (*)Let's hope that the mirror sites respect the monospaced text. I cross my fingers. |