All Questions
Tagged with integer-partitions analytic-number-theory
21
questions
3
votes
1
answer
169
views
How to extend Euler's identity regarding partition on the unit disk?
Theorem (Euler) $:$ For $|x|<1$ we have
$$\prod\limits_{m=1}^{\infty} \frac {1} {1-x^m} = \sum\limits_{n=0}^{\infty} p(n) x^n,$$ where $p(n)$ denotes the number of partitions of $n$ for $...
- 2,013
6
votes
0
answers
181
views
Almost a prime number recurrence relation
For the number of partitions of n into prime parts $a(n)$ it holds
$$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$
where $q(n)$ the sum of all different prime factors of $n$.
Due to https://oeis....
- 13.9k
4
votes
0
answers
229
views
Newman's proof of the Asymptotic Formula for the Partition Function
I'm working on Donald J. Newman's proof that $p(n) \sim \frac{1}{4\sqrt{3}n}e^{\pi\sqrt{\frac{2n}{3}}}$, as found in Chapter II of his book Analytic Number Theory.
Here's what we have so far: the ...
- 727
6
votes
1
answer
228
views
Recently proposed problem by George Andrews on partitions in Mathstudent Journal (India)
Show that the number of parts having odd multiplicities in all partitions of $n$ is equal to difference between the number of odd parts in all partitions of $n$ and the number of even parts in all ...
0
votes
1
answer
248
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Number of pairwise non-isomorphic spanning trees of the wheel $W_n$, with restrictions
I recently encountered this problem. Frankly I'm stuck; would be nice for some help. Here it is:
Let $N,k$ be positive integers. By $p_k(N)$ we denote the number of integer partitions of $N$ with ...
- 2,003
8
votes
2
answers
249
views
Minimizing over partitions $f(\lambda) = \sum \limits_{i = 1}^N |\lambda_i|^4/(\sum \limits_{i = 1}^N |\lambda_i|^2)^2$
I'm trying to characterize the behavior of the the quantity:
$$A = \frac{\sum \limits_{i = 1}^N x_i^4}{(\sum \limits_{i = 1}^N x_i^2)^2},$$
subject to the constraints that
$$ \sum \limits_{i = 1}^N ...
- 664