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Tagged with combinatorics number-theory
1,981
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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/...
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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 $x$.
$$...
- 840
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Generating function of partitions of $n$ in $k$ prime parts.
I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$.
I know ...
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81
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The n-th number open problems
Some open problems in mathematics boil down to the question of defining the $n$-th term of a certain sequence for a specific $n$. For instance, the value of the $5$-th diagonal Ramsey number and the $...
- 1,773
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152
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About the product $\prod_{k=1}^n (1-x^k)$
In this question asked by S. Huntsman, he asks about an expression for the product:
$$\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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214
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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...
$$
\sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}\sum_{q_{3}=q_{1}}^{q_{2}}...\sum_{q_{k+1}=q_{1}}^{q_{k}}x^{q_{1}+q_{2}+...+q_{...
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Counting solution to congruences
I want to count the $x, y \mod a$ and $r, s \mod b$ subject to the following conditions (defining $u, v, w, k$ which exist by the coprimality conditions)
$$(a, x, y) = 1$$
$$(b, r, s) = 1$$
$$ as+xr+...
- 1,285
2
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2
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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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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
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1
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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
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0
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110
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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
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77
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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 ...
- 115
2
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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 \...
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