All Questions
Tagged with combinatorics number-theory
1,985
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Prove that for $n$ sufficiently large and $k \leq n$, there exists $m$ having exactly $k$ divisors $\le n$
Prove that for all sufficiently large positive integers $n$ and a positive integer $k \leq n$, there exists a positive integer $m$ having exactly $k$ divisors in the set $\{1,2, \ldots, n\}$.
here is ...
- 39.7k
12
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1
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Partitioning sets such that the sum of 2 elements is Prime
Given an $n >0$ is it possible to partition the set $\mathcal{P} = \{1,2, \cdots, 2n\}$ into $n$ pairs $(a_{i},b_{i})$ such that $a_{i} + b_{i}$ is a prime?
- 182
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Average cluster size of a nxn matrix
I asked a question about the cluster size inside a vector here. As a result, I finally used the expression $\frac{n}{-k+n+1}$ as the average cluster size, although it´s not proved correct for every ...
- 159
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Mean of probability distribution
I have a probability distribution defined by the following density function:
$f(k,j,n,m)=\frac{(m n)! \mathcal{S}_k^{(j)}}{(m n)^k (m n-j)!}$ (With $\mathcal{S}_k^{(j)}$ being the Stirling number of ...
- 159
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$q$-Pochhammer at root of unity
Are there any identities, papers/studies, posts, etc that go over
$$(\ln\zeta_n^k;q)_{\infty} = \prod_{m=0}^{\infty}(1-\frac{2\pi i k q^m}{n})$$
which is sometimes called the $q$-Pochhammer or quantum ...
- 702
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39
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Counting matrix paths for (n,m>2) matrices
Given a $n\times m$ matrix with $k$ elements inside it, I need to calculate the number of arrangements of those $k$ elements that form at least 1 path from the top to bottom matrix row composed of the ...
- 159
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Fractional part of a sum
Define for $n\in\mathbb{N}$ $$S_n=\sum_{r=0}^{n}\binom{n}{r}^2\left(\sum_{k=1}^{n+r}\frac{1}{k^5}\right)$$
I need to find $\{S_n\}$ for $n$ large where $\{x\}$ denotes the fractional part of $x$.
$$...
- 27
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Summation of n-simplex numbers
Gauss proved that every positive integer is a sum of at most three triangular(2-simplex) numbers. I was thinking of an extension related to n-simplex.
Refer: https://upload.wikimedia.org/wikipedia/...
- 181k
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Cut a number to a random integer between 0 and that number. Keep going until that number is 0. How many cuts do we need?
Start with an integer like n = 100 and set it equal to a uniformaly random integer between [0,n] inclusive. Keep cutting it this way until n = 0. What's the expected value of the number of cuts needed?...
- 29.8k
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Generating function of partitions of $n$ in $k$ prime parts.
I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$.
I know ...
- 47.4k
2
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1
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About the product $\prod_{k=1}^n (1-x^k)$
In this question asked by S. Huntsman, he asks about an expression for the product:
$$\prod_{k=1}^n (1-x^k)$$
Where the first answer made by Mariano Suárez-Álvarez states that given the Pentagonal ...
- 15.8k
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The n-th number open problems
Some open problems in mathematics boil down to the question of defining the $n$-th term of a certain sequence for a specific $n$. For instance, the value of the $5$-th diagonal Ramsey number and the $...
- 583
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814
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Counting numbers smaller than $N$ with exactly $k$ *distinct* prime factors
Using common notation, $\omega(n)$ is the number of distinct prime factors on $n$. Similiarly, $\Omega(n)$ is the number of prime factors of $n$, not necessarily distinct: $120=2^{3}\cdot 3 \cdot 5$ , ...
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Regarding scaling in sumsets
Let $A$ be a finite subset of $\mathbb{N}$. We denote the set $\{a_1 +a_2: a_1, a_2\in A\}$ as $2A$. We call the quantity $\sigma[A]:= |2A|/|A|$ as the doubling constant of $A$, and this constant can ...
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For which integers $m$ does an infinite string of characters $S = c_{1} \cdot c_{2} \cdot c_{3} \cdot c_{4} \cdot c_{5} \ldots$ exist
Question:
For which integers $m$ does an infinite string of characters
$$S = c_{1} \cdot c_{2} \cdot c_{3} \cdot c_{4} \cdot c_{5} \ldots$$
exist such that for all $n \in \mathbb{Z}_{>0}$ there are ...
- 432k
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Partition of a number as the sum of k integers, with repetitions but without counting permutations.
The Hardy-Littlewood circle method (with Vinogradov's improvement) states that given a set $A \subset \mathbb{N}\cup \left \{ 0 \right \} $ and given a natural number $n$, if we consider the sum:
$$f(...
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The sequence $0, 0, 1, 1, 3, 10, 52, 459, 1271, 10094, 63133,...$
Let $a_0$ be a permutation on $\{1, 2, ...,N\}$ (i.e. $a_0 \in S_N$) . For $n \geq 0$:
If $a_n(i+1) \geq a_n(i)$, then $a_{n+1}(i) = a_n(i+1) - a_n(i)$.
Otherwise, $a_{n+1}(i) = a_n(i+1) + a_n(i)$.
$...
- 1,029
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How to prove the following partition related identity?
So I want to show that the following is true, but Iam kidna stuck...
$$
\sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}\sum_{q_{3}=q_{1}}^{q_{2}}...\sum_{q_{k+1}=q_{1}}^{q_{k}}x^{q_{1}+q_{2}+...+q_{...
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Counting solution to congruences
I want to count the $x, y \mod a$ and $r, s \mod b$ subject to the following conditions (defining $u, v, w, k$ which exist by the coprimality conditions)
$$(a, x, y) = 1$$
$$(b, r, s) = 1$$
$$ as+xr+...
- 1,285
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On $s$-additive sequences
For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has ...
- 9,592
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How to explain arithmetic form of surprising equality that connects derangement numbers to non-unity partitions?
$\mathbf{SETUP}$
By rephrasing the question of counting derangements from
"how many permutations are there with no fixed points?"
to
"how many permutations have cycle types that are non-...
1
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2
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102
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Numbers in a Fixed Ratio with Same Multiset of Digits
Recently I'm thinking of a problem of “magic simplifying of fractions”. Specifically, we can simplify $\frac{19}{95}$ to $\frac{1}{5}$ as we can “factor” or “cancel” out the “9”. Now I am consider the ...
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Is it true that $\sum_{k=0}^m\binom{n-k}k$ outputs the $(n+1)$th Fibonacci number, where $m=\frac{n-1}2$ for odd $n$ and $m=\frac n2$ for even $n$?
Does $$\sum_{k=0}^m\binom{n-k}k=F_{n+1}$$ where $m=\left\{\begin{matrix}
\frac{n-1}{2}, \text{for odd} \,n\\
\frac n2, \text{for even} \,n
\end{matrix}\right.$ hold for all positive integers $n$?
...
- 17k
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How to compute the Möbius function
I have no clue how to begin this problem. It involves computing the Möbius inversion function $\mu$. This problem comes from Stanley's $\textit{Enumerative Combinatorics}$, vol 1, problem 70, Chapter ...
- 23.1k
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Percolative process distribution not equivalent to coupon collector problem distribution
I have a process where; given a $n\times 1$ matrix initially empty, an element is inserted in it at a random position, with the possibility of repeating the insertion at a filled cell. Then, after a ...
- 159
2
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1
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Growth of the least $k$ such that $\sigma^k$ has a fixed point for each $\sigma\in S_n$
Let $f(n)$ be the least $k\in \mathbb{N}$ such that $\sigma^k$ has a fixed point for each permutation $\sigma\in S_n$. In light of the cycle decomposition of $\sigma$, $f(n)$ is the least $k\in \...
- 2,407
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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
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Applications, Generalisations and developments of Green-Tao Theorem after 2018
The well-known Green-Tao Theorem is definitely one of the most striking results among different area of Mathematics such as: Number Theory, Combinatorics, Graph Theory, Ergodic Theory,... etc.
https://...
- 2,508
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An optimal order estimation problem for combinatorics background
def:
$\gamma(k)=\frac{\sqrt2^k}{\sqrt\pi}\Gamma(\frac{k+1}{2})$
$m_l$ are integers and $|m_l|<M$, $k_l$ are positive integer and $\sum_{l}k_l=k,\sum_{l}k_lm_l=s$.In other words, decompose $s$ into ...
- 91
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Minimal sum of natural representatives of subgroup of $\mathbb{Z}_p^d$
Let $p$ be a prime and $U$ a subgroup of $\mathbb{Z}_p^d$ generated by $(a_1, \ldots, a_d) \in \mathbb{Z}_p^d$. Are there any theorems concerning the minimum of the sum of the natural representatives ...
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