Saturday, January 22, 2011

Pascal Combinations

Sent: Fri, December 22, 2006 4:30:03 PM
Subject: RE: Request for guidence - discovery of some new sequence patterns and its application in combination theory


When I looked up at Wikepedia and Mathworld
(http://mathworld.wolfram.com/),
I did not find any reference to the polygonal numbers when I discovered
this series around end of Nov but yesterday, I checked again with
Wikepedia and found reference to polygonal numbers.

But never-the-less, I have discovered these 2 formulas (but 1 can be
said to iterative result of well known formula but it gives us some new
understanding). Read the URL for complete process and interpretation
as I found out this.

Summary:

The Combination 2nCn can be also summarized as:

y = N x = N + 1 – y

= ∑ αN * ∑ x - Sum Series

y=0 x=1



αN is for N = 3 is 1, 1, 1, 1, which is 1C0, 2C0, 3C0 and 4C0



for N = 4, it is 1, 2, 3, 4, 5 - Seq, which is 1C1, 2C1, 3C1, 4C1
and 5C1



for N = 5, it is 1, 3, 6, 10, 15, 21 - Sum of Seq, which is 2C2,
3C2, 4C2, 5C2, 6C2, 7C2



for N = 6, it is 1, 4, 10, 20, 35, 56, 84 – Sum of Sum of Seq,
which is 3C3, 4C3, 5C3, 6C3, 7C3, 8C3, and 9C3.



Since we assume a single digit like 2 (for 2C1) to be one dimension
(since there are 1 and 1 combinations), and a series like 3 (1, 1, 1),
2 (1, 1), 1 (1) (for 4C2) to a 2-dimension series, for these cases, we
do not have the formula as it starts with 3 dimension plus.



The Combination 2nCn can be also summarized as:



y = N

= ∑ βN * (N+1 – y) C 1 - Combination Series

y=0



αN is for



for N = 3, it is 1, 2, 3, 4 - Seq, which is 1C1, 2C1, 3C1, 4C1



for N = 4, it is 1, 3, 6, 10, 15, 21 - Sum of Seq – 2C2, 3C2,
4C2, 5C2, 6C2,



for N = 5, it is 1, 4, 10, 20, 35, 56, 84 – Sum of Sum of Seq
– 3C3, 4C3, 5C3, 6C3, 7C3, 8C3



for N = 6, it is 1, 5, 15, 35, 70, 126, 210 - Sum of Sum of Sum
of Seq – 4C4, 5C4, 6C4, 7C4, 8C4, 9C4, 10C4.



The above also satisfies nCr = n-1Cr-1 + n-1Cr, used recursively to
make all combinations of nC1 until r becomes 1 for all combinations.

For m+n Cn, it is in the paper.