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
66
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 ...
- 2,007
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
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
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
419
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
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
1
vote
0
answers
42
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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
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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
0
votes
0
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20
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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 ...
- 623
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 ...
- 5,450
0
votes
2
answers
247
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
4
votes
1
answer
2k
views
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
-1
votes
1
answer
33
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Generating functions (parts equal to $2$) [closed]
What is the generating function for partitions into parts equal to $2, 5$ or $7$?
I know the function of distinct parts, but what if they don't have to be distinct?
- 3
0
votes
2
answers
175
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Generating function for partitions into parts
My question is short and (maybe) simple: What is the generating function for partitions into distinct parts equal to $2, 5$ or $7$?
My idea is to use Euler's theorem: $$\sum p(k)x^k=\prod\frac{1}{1-...
- 185
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, ...
- 185
1
vote
1
answer
906
views
Ordered Integer Partition of fixed size
I'm trying to find how many ways there are to partition n into k partitions with max value s. I've found many solutions that does this for unordered partitions. They use recursion and the say that p(n,...
- 121
2
votes
1
answer
371
views
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
views
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
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0
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115
views
Notation for "factorials" of integer partitions?
Let $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ be a partition of $n $, $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k > 0$.
Is there accepted notation (e.g. in the literature) for ...
- 2,579
1
vote
2
answers
286
views
Simple expression for Partition function, $P(n)$, for $n\le20$
I know there are various forms to express the function $P(n)$, the number of partitions for an integer $n$, e.g. see some of the formulas referenced on Wolfram.
But say I only care about the first ~$...
- 645
6
votes
3
answers
1k
views
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
0
votes
1
answer
155
views
Determine if a partition of integer using given parts exists
Is there a way to determine if a partition of integer $n$ into a given set of integers (represented by a vector $\mathbf{x}$) exists, other than enumerating all partitions of $n$ and checking if one ...
- 391
0
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1
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134
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Proof that $ \sum_{i=1}^k p_i = (n-k) $ where $p_i (n)$ is the number of partitions of n into exactly i parts.
I have to proof that
$p_i(n) = p_i(n − i) + p_{i−1}(n − i) + . . . + p_1(n − i).$
for every $ 1 \le i \le n $, where n is number of n partitions has exactly i parts.
Then I have to calculate $p_5(...
- 123
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
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271
views
Count the number of ways n different-sided dice can add up to a given number
I am trying to find a way to count the number of ways n different-sided dice can add up to a given number.
For example, 2 dice, 4- and 6-sided, can add up to 8 in 3 different ways: ($(2,6),(3,5),(4,4)...
- 1,013