All Questions
Tagged with integer-partitions number-theory
259
questions
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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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0
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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
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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
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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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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
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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
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66
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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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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....
1
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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
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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
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1
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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