All Questions
22
questions
0
votes
1
answer
44
views
The Asymptotic formula of the generating function related with the partition of a positive integer
This question may be duplicate with this answer_1 and here I referred to the same paper by Hardy, G. H.; Ramanujan, S. referred to by wikipedia which is referred to in answer_1.
But here I focused on ...
- 416
3
votes
0
answers
50
views
$\sum_{n=1}^m \frac{1}{n!} \sum_{\sum_{i=1}^n m_i = m, \\ m_i \in \mathbb N_+} \frac{1}{m_1\cdots m_n} = 1$? [duplicate]
I found an equation accidentally when doing my research about branching processes. I think it is correct but I don't know how to prove it:
\begin{equation}
\sum_{n=1}^m \frac{1}{n!} \sum_{\sum_{i=1}^n ...
- 1,972
0
votes
0
answers
41
views
Partitions of integers with finite uses in combinatorics
I've done some research into partitions and am yet to find any resources to understand the following:
Given a number $n$ and the restrictions that:
Using only the numbers $1, 2, ..., m$;
A maximum of ...
- 25
0
votes
1
answer
81
views
Sum of how many numbers should N be partitioned.
Partition of integer:
4 = 4 p(4,1) = 1
= 1+3, 2+2 p(4,2) = 2
= 1+1+2 p(4,3) = 1
= 1+1+1+1 p(4,4) = 1
$max(p(...
user960916
2
votes
1
answer
135
views
Integer partition asymptotics for a finite set of relatively prime integers.
I need to get approximations for partition functions in order to limit the expansion of the generating series used to work out the exact value.
The unrestricted partition function $ p(n) $ counts the ...
- 151
4
votes
3
answers
315
views
Computing Ramanujan asymptotic formula from Rademacher's formula for the partition function
I am trying to derive the Hardy-Ramanujan asymptotic formula
$$p(n) \sim \frac{1}{4n\sqrt{3}}e^{\pi\sqrt{\frac{2n}{3}}}$$
from Radmacher's formula for the partition function $p(n)$ given by
$$p(n)=\...
0
votes
1
answer
55
views
Inequality relying on integer partitions and dominance ordering
Let $\lambda$, $\mu$ be two partitions of a natural number $n$, such that $\lambda$ dominates $\mu$ in the usual dominance order on partitions.
I would like to prove that if $q\geq 2$ is a natural ...
- 488
1
vote
0
answers
121
views
Expressing a sum over the sizes of the parts of every partition of n
Let $(a_1^{r_1},\ldots,a_{p}^{r_{p}})\vdash n$ be the multiplicity representation of an integer partition of n. Each $a_{i}$ is a part of the partition and $r_{i}$ is its corresponding size. We ...
- 1,401
2
votes
1
answer
77
views
Maximizing $\sum\left(\lfloor \frac{n_i}{2} \rfloor+1\right)$ for a partition $\{n_i\}$ of $N$
Let $N$ be a natural number and $\{n_i\}$ be a partition of $N$; by this we mean $1\leq i\leq k$ for some natural number $k$ and $N=n_1+n_2+\cdots+n_k$ where $n_1\geq n_2\geq\ldots\geq n_k\geq1$.
For ...
- 1,107
5
votes
1
answer
231
views
A Conjectured Mathematical Constant For Base-10 Normal Numbers.
Question 1: Let $a$ be a real number with a base-10 decimal
representation $a_1a_2\ldots a_n \ldots$ Denote the number of ways to
write $a_n$ as the sum of positive integers as $p(a_n)$ - also ...
- 3,758
1
vote
1
answer
490
views
Sum over partitions
I want to calculate the following sum over non-negative integer partitions
$$
\sum_{l_1+\cdots +l_n=s} \frac{1}{(l_1!)^2 \cdots (l_n!)^2}.
$$
for fixed $n$ and $s.$
I tried to use Vandermonde's ...
- 321
1
vote
0
answers
49
views
Number of equivalent partitions
This is probably very elementary, but is there a way to get the number of equivalent (not necessarily ordered) partitions of an ordered $k$-fold partition $p=(p_1,..,p_k)$ of $n \in \mathbb{N}$ ...
- 1,460
4
votes
1
answer
188
views
Permutation Partition Counting
Consider the number $n!$ for some integer $n$
In how many ways can $n!$ be expressed as
$$a_1!a_2!\cdots a_n!$$
for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as $\...
- 17.5k
5
votes
1
answer
416
views
Existence of a prime partition
I'm interested in finding out whether there exists a prime partition of a given positive integer $N>1$ such that the partition has specific number of parts.
For instance, as given in another ...
- 51
5
votes
1
answer
108
views
A conjecture on partitions
While trying to prove a result in group theory I came up with the following conjecture on partitions:
Let $b(i,j)$ be the number of partitions of $i$ with greatest part exactly equal to $j$ , for ...
- 10.2k