All Questions
9 questions
12
votes
0
answers
558
views
Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
Definition / Question
Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where
$0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$...
- 19.6k
14
votes
2
answers
1k
views
$n!$ divides a product: Part I
Question. The following is always an integer. Is it not?
$$\frac{(2^n-1)(2^n-2)(2^n-4)(2^n-8)\cdots(2^n-2^{n-1})}{n!}.$$
John Shareshian has supplied a cute proof. I'm encouraged to ask:
...
- 43.2k
11
votes
3
answers
1k
views
A problem on a specific integer partition
Let $n$ be a positive integer, we consider partitions of the following form :
$$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that :
$d_{i}\vert n$
$1=d_{1}<d_{2} \le d_{3} \le ... \le d_{r}$...
5
votes
1
answer
326
views
Generalizing Kasteleyn's formula even more?
Inspired and intrigued by this question, I decided just for fun to throw in another integer into the factors and look what happens. So for $k\in\mathbb Z$, let us define $$K_r(n,k):=\prod_{\ell_1=1}^...
- 13.4k
31
votes
1
answer
2k
views
Navigating $\mathbb{Z}/p\mathbb{Z}$
$\newcommand{\Z}{\mathbb{Z}}$Let's consider a silly-looking question first. Consider $\Z/p\Z$. Say I am allowed the two operations $x\mapsto x+1$ and $x\mapsto 2x$. Then, starting from $0$, I can ...
- 20.2k
11
votes
3
answers
594
views
Is it possible to stab (every rotation of) any four element subset of $\mathbb Z_n$ with less than $n/2$ elements?
Say that $S\subset \mathbb Z_n$ is stabbed by $X\subset \mathbb Z_n$ if for every $t$ we have $(S+t)\cap X\ne \emptyset$.
Is there for every $|S|=4$ an $|X|<n/2$ that stabs it?
My motivation ...
- 18.8k
7
votes
1
answer
569
views
Upper bound for size of subsets of a finite group that contains a sum-full set
Problem
I'm looking for an upper bound for the number $k(G)$ of a finite group $G$, defined as follow:
Let $\mathcal{F}_k$ be the family of subsets of $G$ with size $k$, and we
define $k(G)$ be ...
- 1,250
3
votes
4
answers
654
views
A generalization of Landau's function
For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$
the least common multiple of all ...
- 798
2
votes
0
answers
94
views
Is it possible to stab every permutation of any four element subset of $D_n$ with less than $n/2$ elements?
Say for a permutation group $G$ over $n$ that a set $S\subset \{1,\ldots,n\}$ is G-stabbed by $X\subset \{1,\ldots,n\}$ if for every $g\in G$ we have $gS\cap X\ne \emptyset$.
Is there for every $|S|...
- 18.8k