All Questions
Tagged with integer-partitions integers
29
questions
6
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3
answers
1k
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Integer Partition (Restricted number of parts)
I read an article about integer partition which is posted on Wikipedia, and I found out that the generating function of "partitions of n into exactly k parts" can be represented as:
$$\sum_{n\ge0} ...
- 451
5
votes
2
answers
419
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Maximizing product of three integers
It is well-known that, if we want to partition a positive number $m$ into a sum of two numbers such that their product is maximum, then the optimal partition is $m/2, ~ m/2$. If the parts must be ...
- 10.8k
4
votes
1
answer
2k
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Restricted partition of integer: strictly k distinct parts from a set
I am looking for a way to compute/approach a serie of restricted partitions (or compositions) of integers.
I've found the $Q(n,k)$ quantity (OEIS) satisfying the first two of my following constraints ...
- 43
3
votes
1
answer
120
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Conjugate of conjugate partition
Let $\lambda=(\lambda_1,\dots,\lambda_r)$ is a partition of $n$ and denote $\lambda'$ the conjugate partition of $\lambda$, with $\lambda'_j=\#\{i\,|\,\lambda_i\ge j\}$.
I'm struggling to try to prove ...
- 765
3
votes
1
answer
79
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Sum of integers closest to a given number
I have the following problem at hand:
Given an odd number $n > 7$, find a set of non-negative integers $m_7$, $m_8$, ..., $m_{13}$ and $m_{14}$, such that the sum
$m_7\cdot 7 + m_8\cdot 8 + ... + ...
- 41
3
votes
0
answers
44
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By which scheme should I add the elements in series $(\sum n^{-2})^2$ and $\sum n^{-4}$ to show their rational equivalence?
We know that $\sum n^{-2}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\dots=\frac{\pi^2}{6}$ and $\sum n^{-4}=\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\dots=\frac{\pi^4}{90}$ from "high mathematics" ...
- 253
2
votes
2
answers
90
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How many compositions does 40 have (with up to 5 addends), with each positive integer addend between 2 and 12?
I was on Youtube and found a show called Monopoly Millionaires' Club. I thought it would be interesting to try to calculate the probability of winning the million dollars.
The contestant starts on ...
- 1,562
2
votes
2
answers
174
views
Number of ways to complete a partial Young tableau
Suppose we have a Young tableau with missing entries. Then there can be many number of ways we can complete the Young Tableau.
Is there any specific method to find the number of ways we can complete a ...
- 399
2
votes
1
answer
94
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Divide an integer in the sum of two integers with percentage factor using ceil and floor
I have encoutered a problem in a software that I use for invoicing. I have a variable (quantity) integer A which I want to split in a sum of two integers using a percentage p where $A1 = p*A$ and $A2 =...
- 21
2
votes
1
answer
98
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2-split of $n$ is $\left\{ \lfloor \frac{n}{2} \rfloor,\lceil \frac{n}{2} \rceil \right\}$. What about 3, 4, ...?
Clarification: $k$-split of $n$ is an ordered integer sequence $\left\{ a_1,\cdots,a_k \right\}\quad \text{s.t.}$
$0\le a_1\le\cdots\le a_k$
$a_1+\cdots+a_k=n$
${\left(a_k-a_1\right)}$ is minimized.
...
- 195
2
votes
1
answer
371
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Proof of an integer partitions inequality
I came across an interesting problem the other day.
Let $P_n$ be the number of partitions of a positive integer $n$. For instance $P_4$ = $5$, as there are five ways of partitioning $4$:
$4$
$3+1$
$...
- 233
1
vote
3
answers
128
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Partitioning a sequence of characters into groups of $3$ or $4$
I have some computer code that outputs character sequences of various lengths. For readability, I would like to format output sequences as groups of $3$ or $4$ characters. Examples:
The sequence <...
- 163
1
vote
1
answer
3k
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Coin Change Problem with Fixed Coins
Problem: Given $n$ coin denominations, with $c_1<c_2<c_3<\cdots<c_{n}$ being positive integer numbers, and a number $X$, we want to know whether the number $X$ can be changed by $N$ coins.
...
- 61
1
vote
1
answer
91
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Lower bound on sum of integer vectors
Given 2 integer vectores (add zeros to shortest if necessary) we can sum them term by term to get a new integer vector.
$$ (1,2,3) , (1,5,7) $$
If we add the possibility to permute components from ...
- 623
1
vote
3
answers
2k
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Integer partitions: number of even parts
I've got an elementary, combinatoric question: If the number n is odd, why is the number of even parts = the number of times where each part appears an even number of time = 0 ?
I mean: Of course, ...
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