All Questions
Tagged with co.combinatorics computational-number-theory
28 questions
1
vote
1
answer
207
views
Calculating the value of periodic continued fractions with $a_i\in\lbrace 0,1\rbrace$
Question:
How can the value of continued fractions of the form
$$y:=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\begin{align}\ddots& \\ &a_{n-1}+\cfrac{1}{a_n+y}\end{align}}}}}$$
$$...
- 13.2k
2
votes
1
answer
320
views
Can you confirm the positivity of a quantity involving the Stirling numbers of the first kind
Let $s(m,n)$ denote the Stirling numbers of the first kind. For $m,n\in\mathbb{N}$, define
\begin{equation}
\mathcal{Q}(m,n)=(-1)^n\sum_{\ell=0}^{2n} \binom{m+\ell-1}{m-1} s(m+2n-1,m+\ell-1)\biggl(\...
- 1,091
13
votes
2
answers
741
views
How many players are needed so that two evenly matched teams can be picked?
We have a pool of $n$ players of a game, each player is assigned a "skill" which is an integer $1\leq s\leq 100$. We are now going to pick teams of $5$ players, where the team's skill is ...
- 315
4
votes
2
answers
609
views
Ask for a generating function or an explicit expression of a triangle of positive integers
Preliminaries
I encountered the following triangle of positive integers:
$c_{n,k}$
$n=1$
$n=2$
$n=3$
$n=4$
$n=5$
$n=6$
$n=7$
$n=8$
$k=0$
$1$
$3$
$15$
$105$
$315$
$3465$
$45045$
$45045$
$k=1$
$5$
$...
- 1,091
0
votes
1
answer
97
views
On polynomials associated to integers power sums [closed]
For $0\leq k\leq n$ integers let $P_k(n):= n^k,\ S_k(n):= P_k(1)+\ldots P_k(n)= 1^k+\ldots n^k$.
Then $P_k(0)=0$, $S_0(n)=n$.
For calculate $S_1(n)$ i consider:
$$P_2(n)-P_2(n-1)=2n+1$$
then
$\begin{...
- 4,165
5
votes
0
answers
313
views
A question on infinite arithmetic progressions
I was working on a problem that consisted of deciding if the language a finite automaton (the alphabet of which is $\{0,1\}$ and the words accepted are binary encoded positive integers) contains an ...
- 51
8
votes
1
answer
531
views
How large can the dimension of a 'Span of powers of a finite field basis' be?
Let $p$ be a prime. For finite field $\mathbb{F}_{p^k}$ and $d\in\mathbb{Z}^+$, I am considering the following quantity, where we interpret the field $\mathbb{F}_{p^k}$ also as a $\mathbb{F}_p$-vector ...
- 89
5
votes
0
answers
197
views
Ramsey Numbers for Integers
Erdos defined $f(n)$ to be the minimum $r$ such that there is an $r$-coloring of the positive integers less than $n$, wherein $n$ cannot be written as the sum of distinct monochromatic integers. ...
- 141
11
votes
1
answer
278
views
4-cliques of pythagorean triples graph and its connectivity
Let natural numbers $a, b > 2$ be adjacent if $|a^2 - b^2|$ is a square number. One can find a 3-clique.
For example 153, 185, 697. The questions are: does there exist a 4-clique? Is this graph ...
- 333
5
votes
1
answer
275
views
Is there a short proof for the permutation invariance of this combinatorial map?
Consider a positive integer $n$ and integers $(c_i)_{1\le i \le 4}$, with $1 \le c_i \le n$. Conside the map:
$$f_n: (c_1,c_2,c_3,c_4) \mapsto \delta_{c_1,c_2}\delta_{c_3,c_4} - \# \{ |2n+1-2|x||, \ x ...
0
votes
1
answer
250
views
If the coefficient of the polynomial positive
I want to know what is following sum coefficient looks like. We sum over all integers $p$, $q$ also we put the condition that $q$ is even. Also, it should depend on the parity of $k$
$$\bar{S}(k)=\...
- 685
11
votes
2
answers
410
views
Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$
Theorems V in this paper of L.E. Dickson states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \ \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \...
2
votes
1
answer
199
views
Decide if a system of arithmetic sequences is an $m$-cover of $\mathbb{N}$
Let $A = \{ a_i + b_i \mathbb{N} \}_{i=1}^{k}$, where $a_1, \ldots, a_k \in \mathbb{N} \cup \{0\}$ and $b_1, \ldots, b_k \in \mathbb{N}$ be a system of arithmetic sequences.
For a positive integer $m$...
- 655
6
votes
4
answers
552
views
(Non)uniqueness of the common-factor graph
Let $S=\{x_1,\ldots,x_k\}$ be a set of $k$ distinct natural numbers,
a subset of $\{1,\ldots,n\} = \mathbb{N}_{\le n}$.
Define the common-factor graph $G(S)$ as the (undirected) graph with
a node for ...
- 151k
4
votes
1
answer
755
views
Conjecture on palindromic numbers
The conjecture is as follows:
Let $n\in\mathbb{N}\setminus\{1\}$. Define $a(n)=2^n+1$ and the set:
$$S(n) = \{ (a(n)^m+1)/2\ :\ m\in \mathbb{N}_0\}.$$
Then for all $c\in\mathbb{N}$, the number $(a(n)...
5
votes
1
answer
610
views
Eulerian ordering of the integers modulo n
Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$.
An Eulerian ordering of $C_n$ is an ordering $r_1, \dots, r_n$ of its elements such that:
$$\forall i \le n \ \forall j&...
8
votes
1
answer
484
views
simple conjecture on palindromes in base 10 [closed]
The conjecture says that for any a, b belong to the the set of non-negative integers ($a$ and $b$ are not necessarily distinct), taking any natural value of $c$; we have always that $$(10^c-1) \cdot \...
4
votes
1
answer
254
views
Combinatorial computational problem about 0-1 vectors and sampling algorithms
Let $M \in \{0,1\}^{m\times n}$, where $n\gg 1$ and $m\le n$. A procedure consisting of the following three steps is repeated $t\gg 1$ times:
A row $\require{amsmath} \boldsymbol{r}$ of $M$ picked in ...
- 1,677
6
votes
2
answers
385
views
Is the nth-power-sum graph connected?
This post was inspired by the Square-Sum Problem presented in Numberphile by Matt Parker.
He asked about Hamiltonianness for $n=2$, and we ask about connectedness for all $n \in \mathbb{N}^*$.
...
3
votes
1
answer
288
views
the size of a down-set?
I'm reading a research article lately, and got confused about a question.
So, the fundamental theorem of Kruskal and Katona states that if each set in a given set system $\mathcal{A}$ has $k$ ...
- 131
10
votes
3
answers
1k
views
Explicit formula for elementary symmetric sum
For $k\ge1$, $j\ge1$, Let $$e_k(j)=\sum_{1\le i_1<...<i_k\le j}i_1\cdot\cdot\cdot i_k.$$ We know that $e_k(j)$ is a polynomial in $j$ with coefficients depending on $k$. I am curious about ...
- 113
6
votes
2
answers
461
views
Divisibility labeling on a boolean lattice and positive Euler totient
Let $B_n$ be the rank $n$ boolean lattice (i.e. the subset lattice of $\{1,2, \dots , n \}$). Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum of $B_n$. Let $f: B_n \to \mathbb{N}$ be a ...
7
votes
0
answers
294
views
On the ratio of Gilbreath sequences
Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...
13
votes
1
answer
2k
views
Erdős multiplication problem revisited
This is a well-known problem and is about counting the number of distinct numbers in the $n \times n$ multiplication table.
The very problem has been discussed in-depth and, as such, I require no ...
user56218
1
vote
1
answer
353
views
Valid Difference Sets
Suppose
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K
$$
We calculate the differences as:
$$
d=p_i-p_j\mod N,\quad i\ne j
$$
Now let $a_d$ denote the number of occurrence of $d$ (for $d = 0, 1, 2, ...
- 123
3
votes
2
answers
2k
views
Integer partition and sum of squares
Hello,
The question below might be well known, and using different words (I made these up, I'm not a number theorist or specialist in combinatorics)
For all integers $n\geq 2$ denote by $\mathcal{P}...
- 2,829
0
votes
2
answers
3k
views
Fibonacci Numbers Modulo m [closed]
In the paper "Fibonacci Series Modulo m" by D.D. Wall (found here), there is a table in the Appendix listing values for the function $k(p)$. This function is defined as the period of the Fibonacci ...
- 133
54
votes
2
answers
8k
views
Walsh Fourier transform of the Möbius function
This question is related to this previous question where I asked about ordinary Fourier coefficients.
Special case: is Möbius nearly orthogonal to Morse
August Ferdinand Möbius (November 17, 1790 – ...