Questions tagged [integer-partitions]
Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.
194
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Counting bounded integer solutions to $\sum_ia_ix_i\leqq n$
I want to find the number of nonnegative integer solutions to
$$x_1+x_2+x_3+x_4=22$$
which is also the number of combinations with replacement of $22$ items in $4$ types.
How do I apply stars and bars ...
35
votes
7
answers
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Making Change for a Dollar (and other number partitioning problems)
I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the ...
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89
votes
7
answers
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Number of ways to write n as a sum of k nonnegative integers
How many ways can I write a positive integer $n$ as a sum of $k$ nonnegative integers up to commutativity?
For example, I can write $4$ as $0+0+4$, $0+1+3$, $0+2+2$, and $1+1+2$.
I know how to find ...
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27
votes
1
answer
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The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts
This seems to be a common result. I've been trying to follow the bijective proof of it, which can be found easily online, but the explanations go over my head. It would be wonderful if you could give ...
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17
votes
2
answers
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Partitioning a natural number $n$ in order to get the maximum product sequence of its addends
Suppose we have a natural number $n \ge 0$.
Given natural numbers $\alpha_1,\ldots,\alpha_k$ such that
$k\le n$
$\sum_i \alpha_i = n$
what is the maximum value that $\Pi_i \alpha_i$ can take?
I'm ...
- 271
19
votes
3
answers
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Number of possible combinations of x numbers that sum to y
I want to find out the number of possible combinations of $x$ numbers that sum to $y$. For example, I want to calculate all combination of 5 numbers, which their sum equals to 10.
An asymptotic ...
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15
votes
2
answers
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Identity involving partitions of even and odd parts.
First, denote by $p_E(n)$ be the number of partitions of $n$ with an even number of parts, and let $p_O(n)$ be those with an odd number of parts. Moreover, let $p_{DO}(x)$ be the number of partitions ...
- 153
13
votes
2
answers
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Counting integer partitions of n into exactly k distinct parts size at most M
How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$?
The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at ...
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votes
4
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number of ordered partitions of integer
How to evaluate the number of ordered partitions of the positive integer $ 5 $?
Thanks!
- 5,612
8
votes
1
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A "binomial" generalization of harmonic numbers
For positive integers $s$ and $n$ (let's limit the generality), define
$$H_s(n)=\sum_{k=1}^{n}\frac{1}{k^s},\qquad G_s(n)=\sum_{k=1}^{n}\binom{n}{k}\frac{(-1)^{k-1}}{k^s}.$$
The former is well-known; ...
- 40k
14
votes
6
answers
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How solutions of distinct non-negative solutions are there to $k_1+\cdots+k_n=k$?
How many distinct $n$-tuples with distinct non-negative integer elements are there that add to $k$.
For example there are $6$ triples that add to $4$. Namely $(0, 1, 3)$ and its $6$ permutations. Is ...
- 5,756
2
votes
1
answer
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Number of positive integral solutions of $a+b+c+d+e=20$ such that $a<b<c<d<e$ and $(a,b,c,d,e)$ is distinct
This is from a previous question paper for an entrance exam I am preparing for.
https://www.allen.ac.in/apps/exam-2014/jee-advanced-2014/pdf/JEE-Main-Advanced-P-I-Maths-Paper-with-solution.pdf (Link ...
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2
votes
3
answers
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Multiplication partitioning into k distinct elements
Let's say I have a list with the prime factors of a number $n$ and their corresponding exponents. Is there a formula to calculate how many multiplications with $k$ distinct factors of $n$ are possible?...
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14
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3
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Integer partition of n into k parts recurrence
I was learning integer partition of a number n into k parts(with minimum 1 in each part) and came across this recurrence :
part(n,k) = part(n-1,k-1) + part(n-k,k)
...
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7
votes
3
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Partitions of $n$ into distinct odd and even parts proof
Let $p_\text{odd}(n)$ denote the number of partitions of $n$ into an odd number of parts, and let $p_\text{even}(n)$ denote the number of partitions of $n$ into an even number of parts.
How do I ...
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