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So much for summer, huh?
Back to partitions. It's been hard to find good info on this topic. PlanetMath has some interesting stuff visualizing partitions with Young and Ferrers diagrams--sort of n-ominoes. This looks like a good way to get back to the theme of my sabbatical study...but first I need to understand the theory of partitions more better. http://www.reference.com/browse/wiki/Integer_partition
Scratching too hard at this partition thing will lead you to people like James Sellers, who archives his numerous very erudite articles on the topic: all of which are beyond my level at this point, it seems. So I retreat to http://www.reference.com/browse/wiki/Integer_partition . Remember the recursive function asserted by, but not justified by Dr. Math (he has a master's degree....in math!) a few days ago? Well, the reference.com/wiki site is introduces something a little different they call the "intermediate function:"
For example, For a given k, partitions counted
by - the smallest addend = k
- the smallest addend > k
For our example, two partitionsof 9; that is, 3 The number of partitions meeting the first condition is In our example, we have 2 partitions with minimal addend 3. Now
consider the partitions comprising Obviously, the number of partitions meeting the second condition is
The base cases of this recursive function are as follows: *F*(n, k) = 0 if k > n*F*(n, k) = 1 if k = n
It's helpful to list out a few, to get a feeling for how some patterns vary with the parameters: *F*(4, 1) = 5 : {4, 3-1, 2^{2}, 2-1^{2}, 1^{4}}
*F*(8, 2) = 7 : {8, 6-2, 5-3, 4^{2}, 4-2^{2}, 3^{2}-2, 2^{4}}
*F*(12, 3) = 9 : {12, 9-3, 8-4, 7-5, 6^{2}, 6-3^{2}, 5-4-3, 4^{3}, 3^{4}}
*F*(16, 4) = 11 : (16, 12-4, 11-5, 10-6, 9-7, 8^{2}, 8-4^{2}, 7-5-4, 6^{2}-4, 6-5^{2}, 4^{4})
*F*(20, 5) = 13 : (20, 15-5, 14-6, 13-7, 12-8, 11-9, 10^{2}, 10-5^{2}, 9-6-5, 8-7-5, 8-6^{2}, 7^{2}-6, 5^{3})
*F*(24, 6) = 16 : (24, 18-6, 17-7, 16-8, 15-9, 14-10, 13-11, 12^{2}, 12-6^{2}, 11-7-6, 10-8-6, 10-7^{2}, 9^{2}-6, 9-8-7, 8^{3}, 6^{4})
The simple partition function p(n)
is just http://www.maths.ex.ac.uk/~mwatkins/zeta/partitioning.htm has some interesting connections with physics...also relevant links are |

http://openglut.sourceforge.net/group__bitmapfont.html