All Questions
40
questions
14
votes
0
answers
288
views
All interval sequences mod integers
In music, an all-interval twelve-tone sequence is a sequence that contains a row of 12 distinct notes such that it contains one instance of each interval within the octave, 1 through 11. The more ...
- 2,733
5
votes
0
answers
2k
views
Good books to learn olympiad geometry,number theory, combinatorics and more
I want to start learning olympiad mathematics more seriously, and I would like to have advice on some good books or pdfs to learn with.
I have background but not a big background. For example I know ...
- 2,510
0
votes
0
answers
215
views
Analysis of super-structures emerging in a spiral representation of prime numbers
The fact that each prime number (greater than $9$) ends with one of the four digits $1,3,7,9$, allows us to classify the tens in which the primes are found according to which of these four digits, ...
user559615
10
votes
0
answers
357
views
Dissecting the complexity of prime numbers
Each prime number greater than $9$, written in base $10$, ends with one of the four digits $1,3,7,9$. Therefore, each ten can be classified according to which of these four digits, summed to the ten, ...
user559615
3
votes
1
answer
99
views
Sum of products of $m$-tuples chosen from the set of squared reciprocals
Let $S = \{1/n^2 : n \in \mathbb{N} \}$. We know $\sum S = \zeta(2) = \pi^2/ 6$.
Let $f(S, m)$ be the sum of the products of all $m$-tuples chosen from $S$. That is
$$f(S,m) = \sum_{X \in {S \...
- 10.9k
2
votes
0
answers
49
views
In how many ways can I decompose $n \equiv \pm k \mod{m}$ in a sum of numbers also congruent to $\pm k \mod m$?
I'm trying to quantify the solutions for the following problem:
Given a large $n$ and a small number $m$ such that $n \equiv \pm k \mod{m}$, what is the number of ways I can decompose $n$ on a sum of ...
- 1,307
0
votes
1
answer
505
views
Counting the number of semistandard Young Tableaux with maximum entry $n.$ Reference/Formula request
Question: If $k \leq n$ let $\lambda_k$ be a Young diagram with square $k \times k$ shape. I write $\#_{\lambda_{k}^n}$ to count the
number of semistandard Young tableaux with shape $\lambda_k$ and
...
- 3,758
2
votes
1
answer
304
views
self-conjugate partitions with restrictions
Is there a generating function for the number of symmetric Ferrers diagrams contained in a square ? I think we need to sum up the number self-conjugate partitions of n starting from $n=1$ to $n=k^2$ ...
- 307
4
votes
1
answer
251
views
Novel(?) method of generating Motzkin numbers, Catalan partial sums, and other sequences
For solutions to the equations (1/X)+(1/Y)=K and (X+Y)=1+(1/k) the answer is a pair of points. The X and Y values of a point represent stable points in a two-cycle orbit of the logistic equation ...
- 41
2
votes
1
answer
71
views
Seeking more information regarding the "hybriation function."
Definition 0. Given a pair of finite sets $Y$ and $X$, write $Y_X$ for the set of all collections $\mathcal{K}$ of functions $f : Y \leftarrow X$ that are closed under "hybridization", by which I mean ...
- 68.1k
0
votes
1
answer
54
views
Lattice points in simplices - reference request
I found this paper
http://homepages.math.uic.edu/~yau/35%20publications/An%20upper.pdf
which, in formulas (1.2) and (1.3), relates the number of non-negative and positive integer values that are ...
- 990
10
votes
5
answers
211
views
$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view
A short time ago I found the nice paper Thirty-two Goldbach Variations written by J.M. Borwein and D.M. Bradley. It presents $32$ different proofs of the Euler sum identity
\begin{align*}
\zeta(2,1)=\...
- 109k
2
votes
2
answers
135
views
Finite sums of integers and similar problems: book request
I recently learned about Faulhaber's formula, which says that for each integer $p \geq 1,$ we can simplify the finite sum $\sum_{k \in \mathbb{N}}[k<n]k^p$ so that it becomes an (integer-valued) ...
- 68.1k
1
vote
0
answers
57
views
How to find a set of integer vectors (of length L) such that all its subsets with size L are linearly independent?
Given a number $M\geq L$, how to find a set of $M$ vectors in $\mathbb{Z}_{\geq0}^{L}$, say $S=\{\mathbf{a}_1,\cdots,\mathbf{a}_L\}$, such that:
1-All subsets of $S$ with size $L$ are linearly ...
- 21
4
votes
3
answers
1k
views
Largest subset with no arithmetic progression
I am trying to find some weak bounds on the largest subset of a set, such that the subset has the property that it contains no three elements in arithmetic progression. The elements of the original ...
- 2,242