All Questions
Tagged with combinatorics number-theory
1,983
questions
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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 ...
- 111
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33
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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 ...
- 111
1
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1
answer
37
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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/...
2
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0
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95
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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 $x$.
$$...
- 862
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1
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46
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Generating function of partitions of $n$ in $k$ prime parts.
I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$.
I know ...
0
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0
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82
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The n-th number open problems
Some open problems in mathematics boil down to the question of defining the $n$-th term of a certain sequence for a specific $n$. For instance, the value of the $5$-th diagonal Ramsey number and the $...
- 1,773
2
votes
1
answer
154
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About the product $\prod_{k=1}^n (1-x^k)$
In this question asked by S. Huntsman, he asks about an expression for the product:
$$\prod_{k=1}^n (1-x^k)$$
Where the first answer made by Mariano Suárez-Álvarez states that given the Pentagonal ...
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45
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Partition of a number as the sum of k integers, with repetitions but without counting permutations.
The Hardy-Littlewood circle method (with Vinogradov's improvement) states that given a set $A \subset \mathbb{N}\cup \left \{ 0 \right \} $ and given a natural number $n$, if we consider the sum:
$$f(...
6
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215
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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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77
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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...
$$
\sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}\sum_{q_{3}=q_{1}}^{q_{2}}...\sum_{q_{k+1}=q_{1}}^{q_{k}}x^{q_{1}+q_{2}+...+q_{...
- 1
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0
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38
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Counting solution to congruences
I want to count the $x, y \mod a$ and $r, s \mod b$ subject to the following conditions (defining $u, v, w, k$ which exist by the coprimality conditions)
$$(a, x, y) = 1$$
$$(b, r, s) = 1$$
$$ as+xr+...
- 1,285
2
votes
2
answers
60
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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 ...
- 971
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38
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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-...
3
votes
1
answer
84
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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 ...
2
votes
0
answers
111
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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 ...
- 9,534
1
vote
1
answer
82
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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 ...
- 111
2
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1
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30
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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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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
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75
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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://...
- 2,498
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23
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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
2
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0
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127
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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 ...
- 21
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76
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Let $f$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$. Is $f(k)$ decreasing for all large $k$?
Consider the question
Let $f:\mathbb N \to \mathbb R$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$, $m>1$. Is $f(k)$ decreasing for all $k>k_0$ for some $k_0$?
The answer is clearly no ...
1
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64
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A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
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22
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Exploring the Intersection of Expander Graphs, Number Theory, Representation Theory and Recent Computer Science Developments
I have a solid understanding of the basics of expander graphs and their properties and the recent development of High-Dimensional Expanders and their application to Random Walks, along with other ...
6
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82
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how many natural numbers require at least 6 terms to express as the sum of distinct squares?
I wrote a computer program as an exercise in dynamic programming. It finds the minimum number of distinct squares which sum to some positive target integer n. I found something interesting and would ...
- 261
3
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1
answer
121
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How many $k$-full numbers are there?
Recall that $n\in\mathbb{N}$ is $k$-full if, for a prime number $p$ : $p\mid n$ implies $p^{k}\mid n$. In the paper I am currently reading it is said that there are $O(B^{1/k})$ $k$-full positive ...
- 498
5
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1
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131
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Number of Salem–Spencer subsets of $\{1,2,3,\dots ,n\}$
I was wondering about sets that do not contain any $3$-term AP, and came to know that the official name of such a set is Salem–Spencer set. I was considering the question of counting the number of ...
- 9,534
3
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0
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90
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On thickness of binary polynomials
OEIS A169945 introduces the concept of thickness of a polynomial as the magnitude of the largest coefficient in the expansion of the square of the polynomial. Considering the $2^{n+1}$ polynomials $p(...
- 9,534
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70
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Non negative integral solutions when coefficients are involved and inequality among parameters
What is the number of non negative integral solutions of
$5w+3x+y+z=100$, such that $w≤x≤y≤z$.
I have the answer key to this but I do not understand how to even start. The second inequality condition ...
- 141
8
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1
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263
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Existence of a Subset $S$ of $\mathbb{N}$ Where All but Finitely Many Natural Numbers Are Sums of Consecutive Elements of $S$
I am pondering a question in number theory that touches upon the representation of natural numbers as sums of consecutive elements from a subset S of $\mathbb{N}$. Specifically, the question is:
Does ...
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