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Topic: Singularities in Pascal Triangle.
Replies: 3   Last Post: Jul 2, 2013 7:14 AM

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Angela Richardson

Posts: 42
From: UK
Registered: 6/22/11
Re: Singularities in Pascal Triangle.
Posted: Jul 1, 2013 2:11 AM
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Was there any particular R you were thinking of?

The triangle is all zeroes if 0R0; reflexive relations are boring. Making every other entry 0, i.e. xRy when x(x+1)/2+y  is odd, leaves a 1 at y=ceil(x/2) and 0s elsewhere.  The triangle gets exciting when every third entry is 0, i.e. xRy when x(x+1)/2+y ==2mod3. For example, if y=3k, then the kth top-right-to-bottom-left diagonal is the sequence 2n^k with each member repeated k+1 times, unless k=0 and the diagonal is all 1s.

From: Alin Soare <discussions@mathforum.org>
To: discretemath@mathforum.org
Sent: Sunday, 30 June 2013, 12:52
Subject: Singularities in Pascal Triangle.

I have this problem:

Compute the numbers in the Pascal Triangle, but on some positions (X,Y), the pascal triangle has 0, instead of the sum of P(X, Y-1) + P(X-1, Y-1) , each time when X is in relation R with Y.

Obviously, if I am asked to compute the Pascal Triangle, I do not recursivelly compute the sums for each pair, but I compute the combinations(X,Y).

I wish to ask you whether the methods of analytic combinatorics can help to solve this problem -- to find the values using some analytical formula instead of recurrence.

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