All Questions
Tagged with integer-partitions number-theory
259
questions
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15
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Value of $k$ that gives the highest Restricted-Part Integer Partition Number for $n$
Let $p_k(n)$ be the number of possible partitions of an Integer $n$ into exactly $k$ parts. We know that for any given $n$, $p_k(n)$ gives a non-zero result for $0<k\leq n$, and that the size of ...
1
vote
1
answer
49
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Generating function of partitions of $n$ in $k$ prime parts.
I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$.
I know ...
2
votes
1
answer
155
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About the product $\prod_{k=1}^n (1-x^k)$
In this question asked by S. Huntsman, he asks about an expression for the product:
$$\prod_{k=1}^n (1-x^k)$$
Where the first answer made by Mariano Suárez-Álvarez states that given the Pentagonal ...
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0
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81
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Need help with part of a proof that $p(5n+4)\equiv 0$ mod $5$
Some definitions:
$p(n)$ denotes the number of partitions of $n$.
Let $f(q)$ and $h(q)$ be polynomials in $q$, so $f(q)=\sum_0^\infty a_n q^n$ and $h(q)=\sum_0^\infty b_n q^n$. Then, we say that $f(q)\...
- 350
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77
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How to prove the following partition related identity?
So I want to show that the following is true, but Iam kidna stuck...
$$
\sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}\sum_{q_{3}=q_{1}}^{q_{2}}...\sum_{q_{k+1}=q_{1}}^{q_{k}}x^{q_{1}+q_{2}+...+q_{...
- 1
1
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0
answers
39
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How to explain arithmetic form of surprising equality that connects derangement numbers to non-unity partitions?
$\mathbf{SETUP}$
By rephrasing the question of counting derangements from
"how many permutations are there with no fixed points?"
to
"how many permutations have cycle types that are non-...
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0
answers
24
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Congruences of partition function
I'm trying to understand Ken Ono's results showing Erdös' conjecture for the primes $\ge5$. He first shows the following: let $m\ge5$ be prime and let $k>0$. A positive proportion of the primes $\...
- 73
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0
answers
19
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Estimate the order of restricted number partitions
There are $k$ integers $m_l,1\leq l\leq k
$(maybe negetive), satisfiying $|m_l|\leq M$ and $\sum_l m_l=s$.
I want to get an order estimate of the number of solutions for $k, M$ when fixing $s$.
I came ...
- 91
1
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0
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79
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"Factorization" of the solutions set of a system of linear diophantine equations over non-negative integers
Suppose we have a system of linear diophantine equations over non-negative integers:
$$
\left\lbrace\begin{aligned}
&Ax=b\\
&x\in \mathbb{Z}^n_{\geq0}
\end{aligned}\right.
$$
where $A$ is a ...
- 11
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0
answers
26
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irregularities in partition function modulo n
It is an open problem whether the partition function is even half the time. Inspired by this, I wrote some Sage/Python code to check how many times $p(n)$ hits each residue class:
...
- 387
1
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0
answers
33
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Show the partition function $Q(n,k)$, the number of partitions of $n$ into $k$ distinct parts, is periodic mod m
Let $Q(n,k)$ be the number of partitions of $n$ into $k$ distinct parts.
I want to show that for any $m \geq 1$ there exists $t \geq 1$ such that
$$Q(n+t,i)=Q(n,i) \mod m \quad \forall n>0 \forall ...
- 103
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0
answers
48
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Find generating series on set of descending sequences, with weight function as taking sum of sequence
Given the set of all sequences of length k with descending (not strictly, so $3,3,2,1,0$ is allowed) terms of natural numbers (including $0$), $S_k$, and the weight function $w(x)$ as taking the sum ...
- 183
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45
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Probability that the maximum number of dice with the same face is k
Let say we have $N$ dice with 6 faces. I'm asking my self, what is the probability that the maximum number of dice with the same face is $k$?
In more precise terms, what is the size of this set?
\...
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1
answer
44
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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
1
vote
1
answer
56
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corollary of the partition congruence
I was going though the paper of Ramanujan entitled Some properties of p(n), the number of partitions of n (Proceedings of the Cambridge Philosophical Society, XIX, 1919, 207 – 210). He states he found ...
- 21
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0
answers
32
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Constrained integer partition containing particular summands
Is there a way to calculate the number of constrained integer partitions containing particular summands? By constrained, I mean, the permitted summands must be below a certain limit, such as 5. Take ...
2
votes
1
answer
61
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Counting gap sizes in a subfamily of partitions
Let $\mathcal{OD}$ be the set of all odd and distinct integer partitions. This has a generating function given by
$$\sum_{\lambda\vdash\mathcal{OD}}q^{\vert\lambda\vert}=\prod_{j\geq1}(1+q^{2j-1})$$
...
- 339
1
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24
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Notation for $k$-partitions of $n$ containing at least one summand equal to $s$
I am looking for whether there is any notation for the $k$-partition number of $n$ where the partitions must include some summand $s$.
An example of the kind of notation I am looking for is $P_k^s(n)$....
- 1,129
1
vote
1
answer
57
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What is the 11th unordered combination of natural numbers that add upto 6 in the partition function?
So, I was making unordered combinations of natural numbers which add upto a certain natural number. I was able to go till 6 when I got to know about the partition function. I was pleased to see that ...
- 177
2
votes
1
answer
70
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Identity involving Sum of Inverse of Product of Integer partitions [closed]
Is there a way to prove the following identity
\begin{equation}
\sum_{l = 1}^{k} \left( \frac{(-s)^l}{l!} \sum_{n_1 + n_2 + \ldots n_l = k} \frac{1}{n_1n_2 \ldots n_k} \right)= (-1)^k {s \choose k} \,...
- 89
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0
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36
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Partition of n into k parts with at most m
I ran into a problem in evaluating a sum over kronecker delta. I want to evaluate
$$\sum_{\ell_1,...,\ell_{2m}=1}^s\delta_{\ell_1+\ell_3+...+\ell_{2m-1},\ell_2+\ell_4+...+\ell_{2m}}$$
My approach was ...
- 29
1
vote
1
answer
80
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What is the most precise upper bound for the partitions funcion $p(n)$? [closed]
In the paper "Asymptotic formulæ in combinatory analysis, 1918" Hardy and Ramanujan gave an upper bound $p(n) < \frac{K}{n}e^{2\sqrt{2n}}, K > 0$. Is this the best upper bound? What is ...
- 53
2
votes
0
answers
66
views
Combinatorial explanation for an identity of the partition function
One can employ elementary methods to demonstrate that $p(n) \leq p(n-1) + p(n-2)$ for $n \geq 2$. Recently, I showed that if certain restrictions are imposed on the partitions, the inequality becomes ...
- 907
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0
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39
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MacDonalds Hook content formula derivation
I was trying to understand the derivation of
$$
s_{\lambda}=a_{\lambda+\delta} / a_{\delta}=q^{n(\lambda)} \prod_{x \in \lambda} \frac{1-q^{n+c(x)}}{1-q^{h(x)}}
$$
as given in (https://math.berkeley....
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1
answer
77
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Another formulation for Stirling numbers of the second kind
I find another formulation for Stirling numbers of the second kind:
Let $n\ge k\ge 1$. Denote by
$$\mathbb N_<^n := \{ \alpha = (\alpha_1,\cdots,\alpha_n): 0\le \alpha_1\le\cdots\le\alpha_n, \...
- 1,972
1
vote
1
answer
288
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A certain conjectured criterion for restricted partitions
Given the number of partitions of $n$ into distinct parts $q(n)$, with the following generating function
$\displaystyle\prod_{m=1}^\infty (1+x^m) = \sum_{n=0}^\infty q(n)\,x^n\tag{1a}$
Which may be ...
- 2,813
3
votes
0
answers
50
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$\sum_{n=1}^m \frac{1}{n!} \sum_{\sum_{i=1}^n m_i = m, \\ m_i \in \mathbb N_+} \frac{1}{m_1\cdots m_n} = 1$? [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
3
votes
2
answers
176
views
Where can I find a proof of the asymptotic expresion of partition numbers by Hardy-Ramanujan?
I'm starting to study number theory and I´m interested in partitions, but I don't find a proof of this asymptotic expression $p(n)$ given by Hardy-Ramanujan.
- 39
1
vote
1
answer
83
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Euler's pentagonal number theorem, the notion of $\omega(n)$ and $\omega(-n)$
I'm studying chapter 14 "Partitions" of the famous Apostol's Introduction to Analytic Number Theory. Down at page 311 (section 14.4) and endeavoring to study the pentagonal numbers, Apostol ...
1
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0
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100
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Partitions with maximal length
Suppose $L$ is a non-empty finite set of positive integers, and let $d=\max L$.
For a positive integer $n$, define an $L$-partition of $n$ to be any sequence $a_1,a_2,\ldots,a_m$ of elements of $L$ ...
- 1,136