Question

A partition of a natural number "n" is a representation of "n" as a sum of natural numbers in which order does not matter. So 3+2+2+1+1 is a partition of 9, and 2+1+3+2+1 is the same partition of 9. A number is considered to be a partition of itself, so 9 is a partition of 9. If p(n) means the number of partitions of "n", then the value of p(1)+p(2)+p(3)+p(4)+p(5)+p(6)+p(7)+p(8) is... a) 62 b) 63 c) 64 d) 65 e) 66

Answer 1

Answer:

1

Step-by-step explanation:

2

Answer 2

The partition function $P(n)$ returns the number of partitions for a given number $n$. G. H. Hardy, J. E. Littlewood, and Srinivasa Ramanujan established a precise formula for $P(n)$.

Step-by-step explanation:

Despite the presence of a formula, however, this formula is cumbersome: it is unknown for which values of $n$ the number of divisions of $n$ is even! There is no known simpler formula, and the existence of such a formula is speculative.

Using generating functions to explore partition numbers is a good way to go. $F(x)= sum n geq 0P(n) xn = prod n=1infty frac11-xn$ is the generating function for the sequence $P(n) n geq 0$. The Jacobi theta function, particularly the Jacobi triple product, can be used to investigate partitions.