All Questions
Tagged with integer-partitions power-series
6
questions
2
votes
0
answers
147
views
Could this yield a formula for the Partition numbers?
Background: Lately, I have fallen down the rabbit hole of partition numbers. Specifically the partition function, $p(n)$. It's well known that no closed-form expression (with only finitely many ...
- 4,472
3
votes
0
answers
44
views
By which scheme should I add the elements in series $(\sum n^{-2})^2$ and $\sum n^{-4}$ to show their rational equivalence?
We know that $\sum n^{-2}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\dots=\frac{\pi^2}{6}$ and $\sum n^{-4}=\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\dots=\frac{\pi^4}{90}$ from "high mathematics" ...
- 253
4
votes
1
answer
430
views
How To Apply and Understand the Generating Function for Number Partitioning
The function p(n) counts the number of ways a number can be made up of smaller numbers. For example, the p(5) = 7 because you ...
7
votes
1
answer
233
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What is the significance of this identity relating to partitions?
I was watching a talk given by Prof. Richard Kenyon of Brown University, and I was confused by an equation briefly displayed at the bottom of one slide at 15:05 in the video.
$$1 + x + x^3 + x^6 + \...
- 1,866
0
votes
0
answers
94
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Truncation of partitions generating function question
$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $
$$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of ...
- 1,725
2
votes
1
answer
943
views
Proof that the series for the generating function of the partition function converges?
For $|q| < 1$, the generating function of the partition function $p(n)$ is given by
$$
\sum_{n=0}^\infty p(n) q^n
= \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1}
$$
I have an intuitive ...
- 826