All Questions
Tagged with integer-partitions number-theory
259
questions
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Is the set of all rational additive partitions of a rational number countable?
We usually call additive partitions the set, we call it $P$, of all the ways to write a positive integer $n$ as a sum of positive integers. Formally:
\begin{equation}
P_n = \left\{ (a_1 ,...,a_n)\in\...
3
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Unsolved problems for partition function
In number theory, the partition function $p(n)$ represents the number of possible partitions of a non-negative integer $n$. For instance, $p(4) = 5$ because the integer $4$ has the five partitions $1 +...
- 907
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When are the partition numbers squares?
I'm unsure if this question is even interesting. I am playing around with partition numbers $p(n) :=$ # partitions of $n$, and I noticed that $p(n)$ never really is a square number, except for of ...
- 1,769
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Is there a pattern to the number of unique ways to sum to a number?
I don’t think there is a proper name for these so I will refer to them as “phactors”. Basically, a phactor is a way to sum up to a number using positive real integers that are non zero and not equal ...
- 478
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Representation of number as a sums and differences of natural numbers
Lets consider all the combinations of:
$$1+2+3+4=10,\ \ 1+2+3-4=2,\ \ 1+2-3+4=4,\ \ 1+2-3-4=-4, $$
$$1-2+3+4=6,\ \ 1-2+3-4=-2,\ \ 1-2-3+4=0,\ \ 1-2-3-4=-8,$$
$$-1+2+3+4=8,\ \ -1+2+3-4=0,\ \ -1+2-3+4=2,...
- 1,818
2
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147
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Could this yield a formula for the Partition numbers?
Background: Lately, I have fallen down the rabbit hole of partition numbers. Specifically the partition function, $p(n)$. It's well known that no closed-form expression (with only finitely many ...
- 4,472
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Combinatorial Interpretation of a partition identity
I am working on the book "Number Theory in the Spirit of Ramanujan" by Bruce Berndt.
In Exercise $1.3.7$: He wants us to prove that
$$
np\left(n\right) = \sum_{j = 0}^{n - 1}p\left(j\right)\...
- 439
1
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289
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Number of ways to write a positive integer as the sum of two coprime composites
I've recently learnt that every integer $n>210$ can be written as the sum of two coprime composites.
Similar to the totient function, is there any known function that works out the number of ways ...
- 103
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311
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Partitions of a number for a fixed number of integers
Is there a name for the number of ways to write a positive integer $n$ as a sum of $k$ integers, including 0?
For example, the number 4 can be written as the sum of 3 numbers in the following ways:
4+...
- 1,620
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Show that $p(n-k, k)=p^2(n, k)$
Let $p^m(n, k)$ denote the number of partitions of $n$ having exactly $k$ parts
with each part greater than or equal to $m$. Show that
$p(n-k, k)=p^2(n, k)$, with the convention that $p(n, k)=0$ if $n&...
- 153
1
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3
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73
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Partitions without repetition
I want to know how many partitions without repetition 19 has. I know I should see the coefficient of $x^{19}$ in $$\prod_{k=1}^\infty(1-x^k),$$
but i'm having trouble finding it. Ay hint?
- 153
2
votes
2
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81
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Show that series converges by estimating number of partitions into distinct parts
I need some help with solving the following problem: Let $Q(n)$ be the number of partitions of $n$ into distinct parts. Show that $$\sum_{n=1}^\infty\frac{Q(n)}{2^n}$$ is convergent by estimating $Q(n)...
- 155
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127
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Formula concerning partitions of $n$ and their transpose
I am trying to prove
$$\frac{1}{{n \choose 2}} \sum_j {\lambda_j \choose 2} - {\lambda_j'\choose 2}=\frac{1}{n(n-1)}\sum_j \lambda_j^2 - (2j-1)\lambda_j.$$
Here $\lambda$ is a partition of $n$, i.e. $\...
- 1,092
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2
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How many non-congruent triangles can be formed from $n$ equally spaced vertices on a circle?
How many non-congruent triangles can be formed from $n$ equally spaced vertices on a circle? I tried using the stars and bars method but it isn't working for me. Partitioning $n$ into $3$ parts looks ...
- 11
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Infinite product expression of partition function
I'm working on a problem (specifically, I'm using an exam paper without course notes to prepare for a course starting in September),
Define the partition function $P(q)$ and give its infinite product ...
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