All Questions
Tagged with integer-partitions combinations
52
questions
0
votes
0
answers
19
views
Estimate the order of restricted number partitions
There are $k$ integers $m_l,1\leq l\leq k
$(maybe negetive), satisfiying $|m_l|\leq M$ and $\sum_l m_l=s$.
I want to get an order estimate of the number of solutions for $k, M$ when fixing $s$.
I came ...
- 91
2
votes
3
answers
1k
views
Number of ways of cutting a stick such that the longest portion is of length n
We are given a stick of length $L$ (say). We make cuts such that the longest piece is of length $n$ (say) at most.
What are the minimum number of pieces we will get and in how many ways this can be ...
CommunityBot
- 1
33
votes
5
answers
57k
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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 ...
- 78.4k
0
votes
2
answers
66
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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 ...
- 6,695
7
votes
2
answers
217
views
Number of ways to distribute $n$ identical balls into $k$ identical boxes such that no box contains more than $m$ balls?
I wonder how to count the number of ways (algorithmically is fine) to distribute $n$ identical balls into $k$ identical boxes such that no box contains more than $m$ balls?
I've run into answers in ...
- 62.3k
4
votes
1
answer
3k
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How many ways to write a number $n$ as the product of natural numbers $\geq 2$?
I am looking for a closed form (or efficient algorithm) for $f(n)$, the number of ways in which $n$ can be written as a product of natural numbers $\geq 2$. Note that $f(n)=\sum_{i=1}^{\Omega(n)}{g(n,...
- 535
0
votes
1
answer
277
views
Number of possible combinations of X numbers that sum to Y where the order doesn't matters
I am looking for the number of possible outcomes given to a set of numbers X that sum to Y. This is the same issue as here. However, I would like to consider that (i) the numbers can't be repeated and ...
- 320
0
votes
2
answers
87
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How many tuples of $(a,b,c)$ satisfies the following equation?
When $a,b,c,n\in \mathbb{Z} , a,b,c,n\geq0$ , $a+b+c\leq8n$
How many tuples of $(a,b,c)$ satisfies the following equation?
$a+2b+3c=12n$
I've tried with $n=1$ and there were 13 tuples, but I couldn't ...
- 37.3k
2
votes
2
answers
272
views
Find a bijection between the $(n-1)$ paths and the $n$-paths which have no downramps of even length.
So here is the Question :-
A Dyck $n$-path is a lattice path of n upsteps $(x,y)$ $\rightarrow$ $(x + 1,y + 1)$ and $n$
downsteps $(x,y) \rightarrow (x + 1,y-1)$ that starts at the origin and never ...
- 11
4
votes
0
answers
269
views
Count the number of unique $N \times N$ binary matrices where every two rows or columns can be swapped
Suppose, two $n\times n$ binary matrices are $\it similar$ if one can be transformed to another by swapping any two rows or two columns any number of times.
My problem is: how many unique $n\times n$ ...
- 233
3
votes
3
answers
763
views
Number of non negative integer solutions of $x+y+2z=20$
The number of non negative integer solutions of $x+y+2z=20$ is
Finding coefficient of $x^{20}$ in
$$\begin{align}
&\left(x^0+x^1+\dots+x^{20}\right)^2\left(x^0+x^1+\dots+x^{10}\right)\\
=&\...
- 8,723
1
vote
1
answer
37
views
I want to obtain partition of an integer with an initial value and
I want to obtain a partition of an Integer with an initial value and
the value following it is smaller and the value following it is smaller than the previous value and no value repeats itself.
within ...
- 761
0
votes
0
answers
28
views
Number of partitions with limited cardinality [duplicate]
We are given $k$ urns labeled from $1$ to $k$. What is the number of ways to put $n$ indistinguishable balls into the $k$ (distinct) urns, given that each urn has a limited capacity equal to $c$, ...
- 3,794
1
vote
2
answers
1k
views
The number 3 can be written as $3$, $2+1$, $1+2$ or $1+1+1$ in four ways. In how many ways can the number $n$ be written?
Attempt
Let $x$ be any variable
$X+0=n ; X+Y=n ; X+Y+Z=n ; \dots; X+Y+Z+A+\dots=n$ (sum of n-1 terms); $1+1+1+.......+1=n$ (sum of n terms).
So total number of ways=
$$(n-1) C (1-1)+(n-1) C (2-1)+\...
- 39.9k
0
votes
1
answer
58
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Combination with Restriction and Repetition
I have a number $x$, let's say $5$, and I want to sort the number out into $4$ digits so that the sum of the digits is equal to $5$, but the value of each digit cannot exceed $3$. $0$ would be an ...
- 47.4k