Partition Numbers and Methods for Calculation

The partition as defined by George E. Andrews in his book Number Theory p149 (1971): "A partition of a nonnegative integer n is a representation of n as a sum of positive integers, called summands or parts of the partition. The order of summands is irrevelant." That is, the number of ways a given number can be expressed as the sum of positive integers. For example the three partitions of 3 are 3, 2+1 and 1+1+1 where 2+1 and 1+2 are considered identical. Wikipedia currently has an up to date article on this subject at Partition (number theory).

This page provides programs to calculate partition numbers along with the number of summands using partition programs which were created in 2003 to calculate p(n). One of the Javascript programs, 2003-5h, can be used to calculate for high values of n, viz., 50 and 100.

The output for each p(n) shows a similiarity for the rows for each p(n) after p(15). Below is the output for the p(34) and p(35) partitions where the ellipsis corresponds to the partition numbers p(16) to p(0):

The four Δs are the difference between the numbers in the cell and the closest P(n), viz., n equal to 21, 20, 19, 18 for both the p(34) and p(35). Both of these Δs are repeated from p(16) for every p(n)/p(n+1) pair.

I've found that this program is useful for the construction of Fibonacci type sequences (as a Pascal type triangle construct) and needed a way to calculate each term. The array values of T[i][j] of program 2003-4r2h were modified to produce the desired triangle (see Fibonacci).

Below is a list of the Javascript programs in htm or html and txt format:

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