All Questions
40
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
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
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 ...
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
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 $c$, ...
- 149
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=
$$(n-1) C (1-1)+(n-1) C (2-1)+\...
- 85
2
votes
1
answer
494
views
compositions of n into even parts
I have found here {https://math.stackexchange.com/questions/2167885/compositions-of-n-into-odd-parts} that the number of integer compositions of n into k odd parts would be ${\frac{n+k-1}{2} \choose k-...
- 311
3
votes
3
answers
763
views
Number of non negative integer solutions of $x+y+2z=20$
The number of non negative integer solutions of $x+y+2z=20$ is
Finding coefficient of $x^{20}$ in
$$\begin{align}
&\left(x^0+x^1+\dots+x^{20}\right)^2\left(x^0+x^1+\dots+x^{10}\right)\\
=&\...
- 8,338
0
votes
1
answer
54
views
Applying boundary conditions to counting combinatorial question [duplicate]
I was trying to count the number of natural number solutions to the equation: $x_1 + x_2 + ... + x_{11} = 20$, such that $0 \leq x_i \leq 9$, for all $i \in \{1, ..., 11\}$.
I know how to apply the ...
- 2,663
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 ...
- 314
2
votes
0
answers
63
views
Closed-form solution of sum over compositions?
I am interested in calculating a closed-form solution of the following sum over compositions $$ \sum_{\substack{n_1 + \dots + n_M = N \\ n_i \geq 1}} \dfrac{n_1^2 + \dots + n_M^2}{n_1(N-n_1)! \dots ...
3
votes
2
answers
577
views
How do you find the number of unique parts in a partition of an integer $n$ into $k$ parts?
Suppose I have an integer $n$ and I partition it into $k$ parts. The number of ways this can be done is given by $P(n,k)$, and it satisfies the recurrence relation:
$P(n,k) = P(n-1,k-1) + P(n-k,k)$
...
- 43
1
vote
1
answer
514
views
Combinatorial arguments for number of partitions of $n$ into $k$ distinct parts
Let $Q(n, k)$ be the number of partitions of $n$ into $k$ distinct, unequal parts. Prove $Q(n + {k + 1\choose 2}, k)$ is equal to the number of ways to partition $n$ into at most $k$ parts (parts can ...
user709945
1
vote
1
answer
208
views
Book Recommendations - Discrete Mathematics and Partitions of an Integer
I finished my first discrete math course this semester using mostly the excellent Kenneth Rosen (Discrete Mathematics and Applications) book that was a great help, especially in induction content and ...
- 529
0
votes
0
answers
26
views
Making a group of $p$ people with $n$ available nationalities
Making a group of p people using m out of n available nationalities can be one of these two scenarios;
$m \le p \le n$ or $m \le n \le p$.
Using p,m, and n, how to evaluate the number of ways of ...
- 5,450