All Questions
Tagged with integer-partitions elementary-number-theory
76
questions
6
votes
2
answers
212
views
Partitions of n that generate all numbers smaller than n
Consider a partition $$n=n_1+n_2+\cdots+n_k$$ such that each number $1,\cdots, n$ can be obtained
by adding some of the numbers $n_1,n_2,\cdots,n_k$.
For example, $$9=4+3+1+1,$$
and every number ...
- 415
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
7
votes
1
answer
233
views
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.
$$1 + x + x^3 + x^6 + \...
- 1,866
0
votes
2
answers
101
views
Number of partitions of $n$ formed by combinations of $2$ and $4$
I'm trying to find the number of partitions of a natural number that are a combination of $2$ and $4$.
For example: $$6 = 2+2+2 = 2+4
\Rightarrow p_6 = 2$$
So I start by defining $p_n$ as the ...
- 330
20
votes
1
answer
4k
views
Partition of ${1, 2, ... , n}$ into subsets with equal sums.
The following is one of the old contest problems (22nd All Soviet Union Math Contest, 1988).
Let $m, n, k$ be positive integers such that $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $...
- 383
1
vote
0
answers
77
views
Generating function for writing an even number as a sum of at most k squares
I would like to find the exact number of ways in which $n$ can be represented as a sum of at most $k$ squares such that each term is less than or equal to say, $N$. A generating function for this ...
- 517
-1
votes
1
answer
340
views
How many possibilities of writing a natural number $M$ as a sum of $N$ natural numbers between $0$ and $M$?
How many possibilities are there of writing a natural number $M$ as a sum of $N$ natural numbers between $0$ and $M$?
For example, I need to write $4$, using $4$ numbers between $0$ and $4$. The ...
2
votes
1
answer
114
views
Partitioning a finite set which sums to $n$
Given $n > 1$, we consider the finite sets of positive integers which sum to $n$, and out of these sets we want to maximize the product. For example, given $n = 6$, the set $\{1, 5\}$ does not ...
- 151
1
vote
0
answers
48
views
Proof of the formula for the number of components in all partitions of a given number
I have to show that this formula is the number of components in all partitions of number $n$:
$$\sum_{i=1}^{n}\sum_{j=1}^{[n/i]}\sum_{k=0}^{n-ij}A_i(k) \cdot A_{j-1}(n-ij-k)$$
$A_k(n)$ is number of ...
- 381
2
votes
1
answer
100
views
Explain this generating function
I have a task:
Explain equation:
$$\prod_{n=1}^{\infty}(1+x^nz) = 1 + \sum_{n=m=1}^{\infty}\lambda(n,m)x^nz^m $$
$\lambda(n,m)$ - is number of breakdown $n$ to $m$ different numbers (>0)
It's ...
- 21
4
votes
1
answer
188
views
Permutation Partition Counting
Consider the number $n!$ for some integer $n$
In how many ways can $n!$ be expressed as
$$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
2
votes
0
answers
184
views
Partitions and divisor functions: what is known about their relations?
If $i\geq 1$ is an integer, we have the following integer valued functions (for any integer $n\geq 0$):
\begin{align}
p_i(n)&=\textrm{the number of }i\textrm{-dimensional partitions of }n,\notag\\...
- 14.2k