All Questions
Tagged with integer-partitions integers
29
questions
0
votes
1
answer
54
views
Formula for number of monotonically decreasing sequences of non-negative integers of given length and sum?
What is a formula for number of monotonically decreasing sequences of non-negative integers of given length and sum? For instance, if length k=3 and sum n=5, then these are the 5 sequences that meet ...
- 15
0
votes
2
answers
65
views
Number of positive integral solution of $\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10$
I want to find the number of positive integer solutions of the equations given by
$$\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10.$$
I know the case that, for any pair of ...
- 1,707
2
votes
2
answers
171
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
3
votes
1
answer
120
views
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
views
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
5
votes
2
answers
412
views
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
2
votes
1
answer
93
views
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
96
views
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
1
vote
0
answers
42
views
Algorithm to find the distinct representations of the integer $n$ as a sum of $k$ non-negative p^(th) integer powers.
I am a user of Wolfram Mathematica and in that software there is a function called: PowersRepresentations. This function returns lists of integers $0\le n_1\le n_2\le\dots\le n_k$ such that $n_1^p+n_2^...
- 21
3
votes
0
answers
44
views
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
0
votes
0
answers
20
views
Find minimum integer such that any integer \in [1, n] can be constructed from its consequent subsums
For example, here's (SPOILERS) breakdown for $1143$, which is the solution for $n = 9$
$\underline{1}143$
$\underline{11}43$
$114\underline{3}$
$11\underline{4}3$
$1\underline{14}3$
$\underline{114}3$...
- 230
1
vote
1
answer
91
views
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 ...
- 633
0
votes
2
answers
223
views
Expressing a positive integer as a sum of 3 positive integers
Let us denote the the number of ways in which a positive integer, $n$, can be expressed as a sum of $3$ positive integers (not necessary distinct) by $W_3(n)$.
$W_3(n)$ can be evaluated using any of ...
- 5,468
0
votes
2
answers
245
views
To prove identity $P(n,3)= \lfloor n^2/12 \rfloor$
Suppose $P(n,k)$ is number of partitions of positive integer n by k positive integers with no duplicative tuples. And
$\lfloor r\rfloor$ is largest of integers equal or less than real number $r$
If $...
- 145
2
votes
2
answers
90
views
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