I was wondering if anyone knows if there's a formula to find the number of partitions of a number n. pka?.
I found a formula for the Bell number. Which gives the number of ways of partitioning a set of n objects into subsets.
For instance {1,2,3} has 5 since we have
{1,2,3}, {1,2}{3}, {1,3}{2}, {2,3}{1}, {1}{2}{3}
The formula is \(\displaystyle \L\\\frac{1}{e}\sum_{k=1}^{\infty}\frac{k^{n-1}}{(k-1)!}\)
I found some references to the number of ways to partition a number n into a sum of positive integers, but no formula.
For instance, 5 has 7 partitions:
5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1
Is there a general formula to find the number of ways a number can be partitioned this way?.
I found a clue which says the generating function for the number of partitions of n into distinct integers is \(\displaystyle \L\\(1+x)(1+x^{2})(1+x^{3})..........\). But, What about it?.
I just thought this was interesting to explore.
I found a formula for the Bell number. Which gives the number of ways of partitioning a set of n objects into subsets.
For instance {1,2,3} has 5 since we have
{1,2,3}, {1,2}{3}, {1,3}{2}, {2,3}{1}, {1}{2}{3}
The formula is \(\displaystyle \L\\\frac{1}{e}\sum_{k=1}^{\infty}\frac{k^{n-1}}{(k-1)!}\)
I found some references to the number of ways to partition a number n into a sum of positive integers, but no formula.
For instance, 5 has 7 partitions:
5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1
Is there a general formula to find the number of ways a number can be partitioned this way?.
I found a clue which says the generating function for the number of partitions of n into distinct integers is \(\displaystyle \L\\(1+x)(1+x^{2})(1+x^{3})..........\). But, What about it?.
I just thought this was interesting to explore.
