All Questions
Tagged with integer-partitions combinations
52
questions
0
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0
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19
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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
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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 ...
- 2,007
7
votes
2
answers
217
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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 ...
0
votes
1
answer
277
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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 ...
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 ...
user998872
4
votes
0
answers
269
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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
1
vote
1
answer
37
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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 ...
- 31
0
votes
0
answers
28
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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
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1k
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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=
$$(n-1) C (1-1)+(n-1) C (2-1)+\...
- 85
0
votes
1
answer
58
views
Combination with Restriction and Repetition
I have a number $x$, let's say $5$, and I want to sort the number out into $4$ digits so that the sum of the digits is equal to $5$, but the value of each digit cannot exceed $3$. $0$ would be an ...
- 11
2
votes
1
answer
494
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compositions of n into even parts
I have found here {https://math.stackexchange.com/questions/2167885/compositions-of-n-into-odd-parts} that the number of integer compositions of n into k odd parts would be ${\frac{n+k-1}{2} \choose k-...
- 311
3
votes
3
answers
763
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Number of non negative integer solutions of $x+y+2z=20$
The number of non negative integer solutions of $x+y+2z=20$ is
Finding coefficient of $x^{20}$ in
$$\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
307
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Get combination of numbers that when added same as the given number
For a given number $n >0$ is there a way to get combination that add up to this number??
for example :
if $n=6$ then numbers that add up are $5+1,4+2,3+2+1$ so the combination is 3
if $n=4$ then ...
- 103
0
votes
1
answer
54
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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
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Closed-form solution of sum over compositions?
I am interested in calculating a closed-form solution of the following sum over compositions $$ \sum_{\substack{n_1 + \dots + n_M = N \\ n_i \geq 1}} \dfrac{n_1^2 + \dots + n_M^2}{n_1(N-n_1)! \dots ...
3
votes
2
answers
577
views
How do you find the number of unique parts in a partition of an integer $n$ into $k$ parts?
Suppose I have an integer $n$ and I partition it into $k$ parts. The number of ways this can be done is given by $P(n,k)$, and it satisfies the recurrence relation:
$P(n,k) = P(n-1,k-1) + P(n-k,k)$
...
- 43
1
vote
1
answer
514
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Combinatorial arguments for number of partitions of $n$ into $k$ distinct parts
Let $Q(n, k)$ be the number of partitions of $n$ into $k$ distinct, unequal parts. Prove $Q(n + {k + 1\choose 2}, k)$ is equal to the number of ways to partition $n$ into at most $k$ parts (parts can ...
user709945
1
vote
1
answer
208
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Book Recommendations - Discrete Mathematics and Partitions of an Integer
I finished my first discrete math course this semester using mostly the excellent Kenneth Rosen (Discrete Mathematics and Applications) book that was a great help, especially in induction content and ...
- 529
0
votes
0
answers
26
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Making a group of $p$ people with $n$ available nationalities
Making a group of p people using m out of n available nationalities can be one of these two scenarios;
$m \le p \le n$ or $m \le n \le p$.
Using p,m, and n, how to evaluate the number of ways of ...
- 5,450
0
votes
1
answer
42
views
Integer Partitions of $~n~$ with restrictions.
Provide a generic formula for the number of partitions of an even number $~n~$ where one part has even value and another part also has even value.
Is there some way to approach this problem that uses ...
- 529
2
votes
2
answers
80
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Coefficient of Generating Function
Determine the coefficient of $~x^n~$ in:
$$(x^2 + x^4 + x^6 + ... + x^{n-1})(x + x^3 + x^5 + ... + x^{n-2})$$
Where $~n~$ is an odd number.
How to describe the possible combinations of coefficients ...
- 529
1
vote
1
answer
73
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Number of partitions of $n$ with restrictions
Find the ordinary generating function for the number of partitions of n in which all parts are odd and none surpasses 7. My answer is:
$$\prod\limits_{i=1}^7 \frac{1}{1-x^{2i}}$$
She is correct?
- 529
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1
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70
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Extraction of coefficient from Generating Function with partitions
Determine the coefficient of $~x ^ {15}~$ in:
$(1+𝑥^3+𝑥^6+𝑥^9+𝑥^{12}+𝑥^{15})(1+𝑥^6+𝑥^{12})(1+𝑥^9)$
How to use the fact that the desired coefficient is the number of partitions of 15 in parts ...
- 529
1
vote
1
answer
103
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Partitions of an integer with polynomials
Determine the coefficients of the polynomial $$a_0 + 𝑎_1𝑥_1 + 𝑎_2𝑥_2 + 𝑎_3𝑥_3 + ⋯ + 𝑎_𝑟𝑥_𝑟$$ that has the property that $~𝑎_𝑛 = 𝑝~$ . Where $p$ is the number of partitions of $n$ composed ...
- 529
3
votes
2
answers
241
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Ways of distributing passengers in ships
I need help with the following combinatorial problem. There are $ K $ passengers and $ K $ ships. The passengers are denoted by $ U_1, U_2, \dots, U_K $. The objective is to find in how many ways the $...
- 778
6
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0
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142
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Faa di Bruno's formula and alternating functions
Suppose you have a function $f(x)$ such that ${\rm sgn}\Big(\frac{d^k}{dx^k}\big(f(x)\Big) = (-1)^k$ and a function $g(x)$ such that ${\rm sgn} \Big(\frac{d^k}{dx^k}g(x)\Big) = (-1)^{(k+1)}$, $\forall ...
- 251
0
votes
1
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126
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What is the appropriate weight ($W_k$) (for two arbitrary partitions)?
I already asked a similar question, And from the answer I received, another question came to my mind.
A positive integer can be partitioned, for example, the number 7 can be partitioned into the ...
- 41
3
votes
1
answer
67
views
Is this true for every partitioning?
I have two categories (category1 and category2 ) and The size of both categories is equal to each other. if we partition each categories arbibtrary .Is this proposition proven? or rejected?
$n_T \...
- 41
1
vote
1
answer
125
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How many different ways to pay $2018, using only quarters, dimes, nickels, and pennies?
I have seen solutions that show how this is done for amounts such as $1. Namely I consulted this webpage's explanation--
https://www.maa.org/frank-morgans-math-chat-293-ways-to-make-change-for-a-...
- 11