All Questions
Tagged with integer-partitions number-theory
259
questions
0
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200
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Recurrence relation related to integer partition
I have to prove a recurrence relation which is given by:
$p_{(n,S)} = \displaystyle\sum_{t=0}^{\lfloor \frac{n}{s} \rfloor}$${p_{(n − st , S-\{s\})}}$
assuming that $\{s\}$ is an arbitrary fixed ...
- 997
1
vote
1
answer
57
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Converting between different ways of representing integer partitions
I'm aware of two ways of representing integer partitions, and I'm trying to understand how to write the mapping between then in terms of mathematical notation.
Firstly, we can say that an integer ...
- 1,324
1
vote
1
answer
294
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Minimum number of partitions required for obtaining all numbers from $1$ to $n$
I took part in a coding contest organized by a club in my college for freshers, In one of the problems it was asked to find out the minimum number of partitions of a number $n$, so that all the ...
- 13
1
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0
answers
125
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Generating function of the partition function with distinct parts used to solve infinite product related to bump-functions in topology
I was looking to analytically approximate the integral of a common bump function in topology:
$$ e^{\frac{1}{1-x^{2}}} $$
This problem can be quickly realized into being related to the distinct ...
- 718
2
votes
1
answer
1k
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Number of partitions with x parts
I might be wrong, but this is how I define "parts" when talking about partitions:
The partition 1+1+1 has 3 parts. The partition 1+2+2 has 3 parts. The numbers, whether they be distinct or ...
- 323
1
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0
answers
44
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Partition an integer $n$ into parts so as to maximize the product of the parts [duplicate]
We are given an integer $n\geq 3$. Our goal is to partition $n$ into $k$ parts $p_1, p_2, \ldots, p_k$ with $p_1 + p_2 + \ldots + p_k = n$ (for an arbitrary $k$ we can choose) so as to maximize the ...
- 281
1
vote
1
answer
122
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Number of solutions number theory problem
I am wondering how many nonnegative solutions the following Diophantine equation has: $$x_1+x_2+x_3+\dots+x_n=r$$ if $x_1 \leq x_2 \leq x_3 \leq \dots \leq x_n$
I know if a sequence can be non-...
3
votes
1
answer
85
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Showing $\prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})}$
I want to show
\begin{align}
\prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})}
\end{align}
I know one proof via self-conjugation of partition functions with ...
- 6,490
0
votes
1
answer
116
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There is a way to know how many partitions has a number? [duplicate]
The title says it all, there is a way to know how many partitions has a number?
With "a way" i mean if there is any formula or polynomials to provide the number of ways to "part" a ...
3
votes
1
answer
406
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Computing the Hardy-Ramanujan asymptotic formula using method of steepest descent/saddle point method
I am trying to obtain and prove the Hardy-Ramanujan asymptotic approximation formula given by:
$$p(n) \sim \frac{1}{4n\sqrt{3}}e^{\pi\sqrt{\frac{2n}{3}}},$$
by using Dedekind's eta function
$$\eta(z)=...
4
votes
3
answers
315
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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
2
votes
0
answers
95
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Power series of the form $r \bmod q$
I am trying exercises of Apostol Introduction to analytic number theory and I am struck on this problem of chapter partitions on page 14.
I am struck in part (b) as I have no idea on how to deal with ...
user775699
1
vote
1
answer
124
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Question 14.12 Tom Apostol Introduction to Analytic number theory
I am studying Ch -14 from Apostol's book and could not solve this particular problem.
It's image:
I am unable to Solve 12(a) (I have done (b) ).
As q is prime, so (n, q) =1 or q| n and 11 (b) will be ...
user775699
1
vote
1
answer
62
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Deriving a partition identity from some given identities
I am trying questions from Apostol Introduction to ANT of Chapter partitions and need help in deducing this identity.
Problem is question 6(a) which will use some information from 2 and 5(b).
...
user775699