All Questions
Tagged with combinatorics number-theory
1,983
questions
0
votes
0
answers
32
views
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
1
vote
0
answers
71
views
+50
$q$-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 ...
- 558
0
votes
0
answers
33
views
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
2
votes
0
answers
95
views
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 $x$.
$$...
- 27
1
vote
1
answer
37
views
Summation of n-simplex numbers
Gauss proved that every positive integer is a sum of at most three triangular(2-simplex) numbers. I was thinking of an extension related to n-simplex.
Refer: https://upload.wikimedia.org/wikipedia/...
- 181k
17
votes
2
answers
942
views
Cut a number to a random integer between 0 and that number. Keep going until that number is 0. How many cuts do we need?
Start with an integer like n = 100 and set it equal to a uniformaly random integer between [0,n] inclusive. Keep cutting it this way until n = 0. What's the expected value of the number of cuts needed?...
- 29.8k
1
vote
1
answer
46
views
Generating function of partitions of $n$ in $k$ prime parts.
I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$.
I know ...
- 47.3k
2
votes
1
answer
154
views
About the product $\prod_{k=1}^n (1-x^k)$
In this question asked by S. Huntsman, he asks about an expression for the product:
$$\prod_{k=1}^n (1-x^k)$$
Where the first answer made by Mariano Suárez-Álvarez states that given the Pentagonal ...
- 15.9k
0
votes
0
answers
82
views
The n-th number open problems
Some open problems in mathematics boil down to the question of defining the $n$-th term of a certain sequence for a specific $n$. For instance, the value of the $5$-th diagonal Ramsey number and the $...
- 575
4
votes
4
answers
804
views
Counting numbers smaller than $N$ with exactly $k$ *distinct* prime factors
Using common notation, $\omega(n)$ is the number of distinct prime factors on $n$. Similiarly, $\Omega(n)$ is the number of prime factors of $n$, not necessarily distinct: $120=2^{3}\cdot 3 \cdot 5$ , ...
2
votes
2
answers
60
views
Regarding scaling in sumsets
Let $A$ be a finite subset of $\mathbb{N}$. We denote the set $\{a_1 +a_2: a_1, a_2\in A\}$ as $2A$. We call the quantity $\sigma[A]:= |2A|/|A|$ as the doubling constant of $A$, and this constant can ...
- 971
3
votes
1
answer
84
views
For which integers $m$ does an infinite string of characters $S = c_{1} \cdot c_{2} \cdot c_{3} \cdot c_{4} \cdot c_{5} \ldots$ exist
Question:
For which integers $m$ does an infinite string of characters
$$S = c_{1} \cdot c_{2} \cdot c_{3} \cdot c_{4} \cdot c_{5} \ldots$$
exist such that for all $n \in \mathbb{Z}_{>0}$ there are ...
- 429k
0
votes
0
answers
45
views
Partition of a number as the sum of k integers, with repetitions but without counting permutations.
The Hardy-Littlewood circle method (with Vinogradov's improvement) states that given a set $A \subset \mathbb{N}\cup \left \{ 0 \right \} $ and given a natural number $n$, if we consider the sum:
$$f(...
6
votes
0
answers
215
views
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
0
votes
0
answers
77
views
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