All Questions
32
questions
1
vote
1
answer
101
views
On $(0,1)$-strings and counting
Consider a binary string of length $n$ that starts with a $1$ and ends in a $0$. Clearly there are $2^{n-2}$ such bit strings. I would like to condition these sequences by insisting that the number of ...
- 339
1
vote
1
answer
44
views
Graded ring generated by finitely many homogeneous elements of positive degree has Veronese subring finitely generated in degree one
Let $S=\bigoplus_{k\ge 0}S_n$ be a graded ring which is generated over $S_0$ by some homogeneous elements $f_1,\dotsc, f_r$ of degrees $d_1,\dotsc, d_r\ge 1$, respectively. I want to show that there ...
- 620
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
19
votes
1
answer
1k
views
Can we partition the reciprocals of $\mathbb{N}$ so that each sum equals 1
Let $S = \{1, 1/2,1/3,\dots\}$
Can we find a partition $P$ of $S$ so that each part sums to 1, e.g.
$$P_1 = {1}$$
$$P_2 = { 1/2,1/5,1/7,1/10,1/14,1/70}$$
$$P_3 = {1/3,1/4,1/6,1/9,1/12,1/18}$$
$$P_4 = \...
- 702
3
votes
0
answers
50
views
$\sum_{n=1}^m \frac{1}{n!} \sum_{\sum_{i=1}^n m_i = m, \\ m_i \in \mathbb N_+} \frac{1}{m_1\cdots m_n} = 1$? [duplicate]
I found an equation accidentally when doing my research about branching processes. I think it is correct but I don't know how to prove it:
\begin{equation}
\sum_{n=1}^m \frac{1}{n!} \sum_{\sum_{i=1}^n ...
- 1,972
1
vote
1
answer
161
views
How to make this proof rigorous by introducing partition of numbers?
Question: Let $n$ be a positive integer and $ H_n=\{A=(a_{ij})_{n×n}\in M_n(K) : a_{ij}=a_{rs} \text{ whenever }
i +j=r+s\}$. Then what is $\dim H_n$?
Proof:
For $n=2$
$H_2=\begin{pmatrix} a_{11}&...
- 13k
-3
votes
4
answers
129
views
Find a minimal set whose elements determine explicitly all integer solutions to $x + y + z = 2n$
Is there a way to exactly parameterise all the solutions to the equation $x + y + z = 2n$, for $z$ less than or equal to $y$, less than or equal to $x$, for positive integers $x,y,z$?
For example, for ...
- 67
0
votes
0
answers
41
views
Partitions of integers with finite uses in combinatorics
I've done some research into partitions and am yet to find any resources to understand the following:
Given a number $n$ and the restrictions that:
Using only the numbers $1, 2, ..., m$;
A maximum of ...
- 25
1
vote
0
answers
62
views
Number of solutions to linear equation $x_1+x_2+\dots+x_n=m$ when the domain of $x_i\ne$ domain of $x_j$
In the lecture notes of one of my previous classes, it was used that if we have an equation of the form
$$\tag{1}
x_1+x_2+\dots+x_n=m
$$
then the total number of solutions, when each $x_i$ is a non-...
- 175
1
vote
3
answers
141
views
Partitioning into products
Consider partitions where every summand has a factor in common with its neighbors and only $x_n$ can be one:
$$x_0 x_1 + x_1 x_2 + x_2 x_3 + \cdots + x_{n-1} x_n = N \qquad x_i \in \mathbb{N}$$
...
- 2,125
1
vote
2
answers
119
views
Unique ways to write $n$ as sum of three distinct nonnegative integers up to the order of the summands
How many ways are there to express a natural number, $n$, as the sum of three whole numbers, $a,b,c$, where $a,b,c$ are allowed to be 0 but are unique?
For example: $n=9$ there are only seven ways: $1+...
- 64
4
votes
3
answers
315
views
Computing Ramanujan asymptotic formula from Rademacher's formula for the partition function
I am trying to derive the Hardy-Ramanujan asymptotic formula
$$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)=\...
1
vote
2
answers
398
views
A question about integer partitions
Lets say that $p(n)$ is the number of ways of partitioning $n$ into integers (order doesn't matter).
How does one prove that $$p(n) \equiv p(n|\text{distinct odd parts}) \mod 2$$?
For $n=1$, there's ...
- 5,215
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
17
votes
0
answers
255
views
For what $n$ can $\{1, 2,\ldots, n\}$ be partitioned into equal-sized sets $A$, $B$ such that $\sum_{k\in A}k^p=\sum_{k\in B}k^p$ for $p=1, 2, 3$?
This is a recent problem in American Mathematical Monthly. The deadline for this question just passed:
$\textbf{Problem:}$ For which positive integers $n$ can $\{1,2,3,...,n\}$ be partitioned into ...
- 1,914