All Questions
Tagged with integer-partitions combinations
52
questions
-1
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1
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69
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Total Collections of integers that sum to constant
For a range of positive integers $1 - S$, how many collections of $N$ integers are there that their sum is a constant $S$.
Example:
Integers from $1$ to $100$
Collections of $4$ integers
Each ...
- 123
2
votes
1
answer
4k
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Number of positive integral solutions of $a+b+c+d+e=20$ such that $a<b<c<d<e$ and $(a,b,c,d,e)$ is distinct
This is from a previous question paper for an entrance exam I am preparing for.
https://www.allen.ac.in/apps/exam-2014/jee-advanced-2014/pdf/JEE-Main-Advanced-P-I-Maths-Paper-with-solution.pdf (Link ...
- 53
0
votes
0
answers
70
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Number of ways to partition $\{1,2,3, \dots, N\}$ into tuples where the size of no tuple exceeds $3$.
While it seems to me that the general answer is not going to be a neat formula, I really only need this for $N=4$ and $N=5$. I'm getting $61$ and $321$ respectively, but I'm not sure. Please help.
- 981
1
vote
1
answer
951
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Coin Combinations for any given scenario.
I am trying to work out the number of scenarios I can cover with a given set of coin combinations so I can decide when I have the optimal amount of change to carry.
For the sake of the example, lets ...
- 325
6
votes
4
answers
672
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How many times does $k$ occur in the composition of $n$?
How many times does the number $k$ occur in the composition of $n$?
Composition of Integer
In short, the difference between the partition of an integer and composition is the order of numbers. In ...
- 237
1
vote
1
answer
92
views
Integer composition in exactly $T$ parts with maximum addend constraint.
In how many ways an integer $N$ can be partitioned into exactly $T$ parts such that
$T = \lfloor N/K \rfloor + 1$
$N = A_1 + A_2 + \cdots+ A_T$ where order matters
$0 \lt A_i \leq K$
$ N \bmod K \...
- 237
2
votes
3
answers
1k
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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 ...
1
vote
0
answers
157
views
Ways to arrange in a two dimensional array an increasing sequence
Given a $n\times m$ grid, which has the number 1 in the upper-left square and a positive integer $1\leq k\leq n+m-1$ in the lower right-square, I am trying to determine in how many ways can the ...
- 11
0
votes
0
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465
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Restricted Composition
I am trying to find the number of compositions of a given number restricted by the numbers present in a subset.
I read that this is called A-Restricted Composition. where A is a set of numbers
...
0
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2
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240
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Number of solutions using partitions for linear equation having restrictions
Here is a linear equation $$a+b+c+d=12$$ where $a,b,c,d$ are restricted to be greater than zero and less than or equal to 6.
How many set of positive integer solutions are possible using partitions ...
5
votes
1
answer
4k
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How many permutations in S(n) have this particular type?
I'm working through the textbook A Course in Enumeration. In the section about permutations and Stirling numbers, I'm having trouble understanding problem 1.45. It is:
We fix $n \in \mathbb{N}$, and ...
- 468
1
vote
0
answers
52
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Relation of relative numbers of (restricted) ways to distribute identical / distinct objects into distinct bins
If want to know if the following inequality holds for general values of $s \leq n \ll m$.
$$\frac{C_0(n,m,s)}{C_0(n,m)} \leq \frac{p(n,m,s)}{m^n}$$
$C_0(n,m) = \binom{n+m-1}{m-1}$ is the number of ...
- 63
0
votes
1
answer
82
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Partitioning a Queue
Imaging a situation that we have n people in a queue and each people represent with number 1 and I want to partition the queue in smaller part, there are several ways to partition the queue.
For ...
- 135
3
votes
0
answers
112
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Amount of combinations of sets summing to number
(Apologies for the confused arbitrariness here; I don't have experience in formal maths to make abstract my lay-person thoughts, but I've tried my best.)
I have $x$ identical but order-important sets ...
- 31
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,...
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