All Questions
133
questions
0
votes
3
answers
111
views
The number of integer solution $x_1+x_2+\cdots+x_k=n$ such that $x_i\in \{1,2\}$
I am trying to find a combinatorial interpretation for the number of integer solution of the equation
$$x_1+x_2+\cdots+x_k=n$$ such that $x_i\in \{1,2\}$.
I know that the number of this solution is ${...
- 649
1
vote
1
answer
49
views
How to count the total unique numbers in this set?
If we have the following set:
$\Big\{ \frac{1}{1},\frac{2}{1},...,\frac{k}{1},\frac{1}{2},...,\frac{k}{2},...,\frac{1}{k},...,\frac{k}{k} \Big\}$
it is very clear that there are some doublets of ...
0
votes
1
answer
54
views
$P_{n,k} = \frac{S_{n,k}}{k!}.$ Partiton of integers and surjective functions.
Let $S_{n,k}$ denote the number of surjective functions from $[n] \to [k]$ and $P_{n,k}$ denote the number of partitions (integer) of $n$ into $k$ parts. To prove by combinatorial argument that $$P_{n,...
- 12.7k
1
vote
0
answers
216
views
A $10$ digit number with distinct digits such that the following holds:
A $10$ digit number with distinct digits is given and using all of its digits two new numbers are created. The sum of the two new numbers is $99999$ and their product is the same as the $10$ digit ...
- 12.7k
2
votes
1
answer
62
views
Question about coefficients of generating functions
Theorem: Let $n> 0 \in \mathbb Z.$ Let $p_n$ stand for the number of integer partitions of $n$ and let $k$ be the number of consecutive integers in a partition. Then $p_n + \sum_{k \ge 1}(-1)^k(p_{...
- 23
2
votes
1
answer
165
views
About the proof of Euler’s Pentagonal Number Theorem on Wiki
Euler’s Pentagonal Number Theorem on Wikipedia
For convenience, here below is the statement:
Let $n$ be a nonnegative integer, let $q_e(n)$ be the number of partitions of $n$ into even number of ...
- 23
7
votes
1
answer
344
views
Combinatorics With Relations
The twelvefold way offers a framework for counting functions, under various conditions which can be expressed as n-fold Cartesian Products of the function's domain, function, and codomain attributes. ...
- 1,922
1
vote
0
answers
78
views
Is the number of sub-boolean algebra of a set with size n , Bell(n)?
In boolean algebra (P(S),+,.,’) we must have S as 1 and {} as 0 in every possible sub-boolean algebra to hold id elements.
We must have S-x for every subset x⊆S to hold complements.
It seems like ...
- 313
1
vote
1
answer
187
views
Can we use division algorithm here? $a = 17$ and $b = −3$
But I think is that we can use it where reminder is $2$ and quotient is $-5$. But I am confused about quotient is this ok for quotient to be a negative number? We are finding $\space b \mid a \space$ ...
- 119
0
votes
1
answer
119
views
number of combinations of integers that satisfy equation
I am aware the number of distinct non-negative integer-valued vectors $(x_1, x_2, ..., x_r)$ satisfying the equation $x_1 + x_2 + ... + x_r = n$ is given by
$$n+r-1 \choose r-1$$
However, is there a ...
- 6,099
9
votes
1
answer
899
views
Expected Value for the Number of Parts of a Random Partition (Considering Only a Portion of the Partition Spectrum)
Let $n$ be a positive integer. If we take the set of all partitions of $n$ and choose a random partition from it (uniformly), then the expected value of the number of parts of this partition is a ...
- 113
3
votes
0
answers
43
views
Finite generation of a subset of an algebraic structure.
Consider the set $\mathbb{N}^n$ of $n$ tuples of positive integers, with the following partially defined "operations" $\oplus_i$.
This operation takes in two vectors $A=(a_1,..a_i,...a_n)$, $B=(b_1,.....
- 6,900
5
votes
0
answers
278
views
Expected Value for the Number of Parts of a Partition of n
Given a positive integer $n$, I want to know the expected value for the number of parts of a random partition of $n$.
I am aware that a similar question has been asked already: Expected number of ...
- 113
5
votes
4
answers
1k
views
How to find number of integers not divisible by 2 nor 3 in a range?
I'm trying to come up with a formula to find the number of numbers in a range that are neither divisible by 2 nor 3. For example between 20 and 30 their are 3, namely 23, 25 and 29.
I had the formula ...
- 449
2
votes
3
answers
347
views
Number of solutions pairs of $ax+by=n$
Question: Given that $a, b, n$ are positive integers and $x, y$ are nonnegative integers such that
$$ax+by=n$$
has at least one solution pair $(x, y) $.
How many solution pairs $(x, y) $ are ...
- 15.9k