All Questions
Tagged with integer-partitions elementary-number-theory
76
questions
1
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1
answer
101
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On $(0,1)$-strings and counting
Consider a binary string of length $n$ that starts with a $1$ and ends in a $0$. Clearly there are $2^{n-2}$ such bit strings. I would like to condition these sequences by insisting that the number of ...
- 339
1
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1
answer
62
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Computing integer partitions subject to certain constraints
Context:
I am programming a videogame.
Background:
Let $I$ be a set of named items such that each is assigned a difficulty score, and each is tagged either as "food" or "obstacle". ...
- 1,312
1
vote
1
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44
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Graded ring generated by finitely many homogeneous elements of positive degree has Veronese subring finitely generated in degree one
Let $S=\bigoplus_{k\ge 0}S_n$ be a graded ring which is generated over $S_0$ by some homogeneous elements $f_1,\dotsc, f_r$ of degrees $d_1,\dotsc, d_r\ge 1$, respectively. I want to show that there ...
- 620
0
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1
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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
19
votes
1
answer
1k
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Can we partition the reciprocals of $\mathbb{N}$ so that each sum equals 1
Let $S = \{1, 1/2,1/3,\dots\}$
Can we find a partition $P$ of $S$ so that each part sums to 1, e.g.
$$P_1 = {1}$$
$$P_2 = { 1/2,1/5,1/7,1/10,1/14,1/70}$$
$$P_3 = {1/3,1/4,1/6,1/9,1/12,1/18}$$
$$P_4 = \...
- 702
6
votes
1
answer
83
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Partition of a number $n$ whose each part is coprime with this number
I'm trying to solve the following problem: given an integer $n$, under which conditions of $n$ the following statement is true:
For any $1 < k \leq n$, there is always a partition of $k$ parts of $...
- 598
0
votes
0
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73
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Factoring an integer $N$ using its random partition of length $3$
While working on this MSE question that I had posted, I wondered what would be a minimal base of numbers that we could work with the algorithm described in that question.
I conjectured that a ...
- 3,341
3
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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
0
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0
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60
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Books for developing an intuitive understanding of the partitions of numbers
I understand from the fundamental theorem of arithmetic that every number can be written as a product of its prime factors,but I’m curious about partitions,how numbers can be broken up into sums and ...
0
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1
answer
33
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Can I always split a friendly partition into two subpartitions, one of which is composed solely of 1, 2, 3, 4, and the largest part?
Suppose I have a partition $p$ of a positive integer $n$, $p$ is defined to be a friendly partition if and only if the following hold:
$n$ is divisible by $4$.
The length of $p$ is $\frac{n}{2}$.
...
- 11.8k
1
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1
answer
161
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How to make this proof rigorous by introducing partition of numbers?
Question: Let $n$ be a positive integer and $ H_n=\{A=(a_{ij})_{n×n}\in M_n(K) : a_{ij}=a_{rs} \text{ whenever }
i +j=r+s\}$. Then what is $\dim H_n$?
Proof:
For $n=2$
$H_2=\begin{pmatrix} a_{11}&...
- 13k
-3
votes
4
answers
129
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Find a minimal set whose elements determine explicitly all integer solutions to $x + y + z = 2n$
Is there a way to exactly parameterise all the solutions to the equation $x + y + z = 2n$, for $z$ less than or equal to $y$, less than or equal to $x$, for positive integers $x,y,z$?
For example, for ...
- 67
2
votes
0
answers
214
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Maximize sum of ceiling functions
I need to find the maximum of a sum of ceiling functions. The following are given
$$N,C\in\mathbb{Z}\text{ with }0\leq N\text{ and }1\leq C$$
$$\frac{p}{q}\in\mathbb{Q}\text{ with }p,q \text{ coprime ...
- 87
0
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0
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41
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Partitions of integers with finite uses in combinatorics
I've done some research into partitions and am yet to find any resources to understand the following:
Given a number $n$ and the restrictions that:
Using only the numbers $1, 2, ..., m$;
A maximum of ...
- 25
1
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0
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62
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Number of solutions to linear equation $x_1+x_2+\dots+x_n=m$ when the domain of $x_i\ne$ domain of $x_j$
In the lecture notes of one of my previous classes, it was used that if we have an equation of the form
$$\tag{1}
x_1+x_2+\dots+x_n=m
$$
then the total number of solutions, when each $x_i$ is a non-...
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