All Questions
Tagged with combinatorial-group-theory co.combinatorics
12 questions
2
votes
2
answers
309
views
Combinatorial problem in $G(32, \, 6)$
The following problem arose when studying the same type of questions in Algebraic Geometry that led me to my previous question MO379272.
Let us consider the group $G$ of order $32$ whose label in GAP4 ...
- 66.3k
4
votes
1
answer
166
views
Complementing the red and blue boolean cube?
Given a boolean $0/1$ cube in $n$ dimensions with $2^{n-1}$ red and $2^{n-1}$ blue points can we complement the cube (blue becomes red and vice versa) in $\operatorname{poly}(n)$ transformations?
...
- 13.9k
8
votes
1
answer
248
views
Number of words of length N that reduce to the identity in a specific Coxeter group
Suppose we have a Coxeter group whose diagram is given by a simplex. In other words, $G=\langle g_1,\ldots ,g_k\mid(g_i)^2=e,\,(g_ig_j)^3=e \rangle$. How many words of length $N$ simplify to the ...
- 525
6
votes
0
answers
479
views
Darkness in the lamplighter group
Consider paths through the lamplighter group $\mathbb{Z}_n\wr\mathbb{Z}$ with steps consisting of moving left, moving right, and toggling the lamp at the current position. How many paths of length $m$ ...
- 2,203
6
votes
1
answer
441
views
Minimum number of operations necessary to arrive at any configuration
Let $k \geq 2$ and $N_1, N_2, ..., N_k$ be positive integers.
Let $S=\{(a_1,a_2,...,a_k) \in \mathbb{Z}^k:1 \leq a_i \leq N_i\}$ and $A=\{1,2,...,\prod_{i=1}^{k} N_{i}\}$.
Given a bijective map $f:...
- 3,153
12
votes
1
answer
415
views
"Bisecting" a free subgroup with respect to word length
My broad question is regarding the lengths of (reduced) words in a subgroup of a free group.
As motivation, consider the free group $Gp(S)$ where $|S|=n$, that is, a free group of rank $n$. Let $S=\{...
- 949
23
votes
1
answer
1k
views
Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$
When I was too young one of my problems was in the list of problems of All-Russian Olympiad. The problem is the following:
Problem. We have a surface of a cube $n\times n \times n$ such that each ...
- 508
3
votes
3
answers
129
views
Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)
Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray*}
s_0 &=& a, \\
s_1 &=& b, \text{and} \\
s_{n+2} &=& s_{n+1} ...
user76098
0
votes
1
answer
71
views
Monotonicity of the gap of permutated sequence
Let $a$ be an arbitrary sequence and denote by $\mbox{gap}_k(a) = a_{(k)} - a_{(k+1)}$, where $a_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of $...
- 773
3
votes
0
answers
5k
views
How many combinations does Android pattern have? [closed]
Rules-
1) At-least 4 and at-max 9 dots must be connected.
2) There can be no jumps
3) Once a dot is crossed, you can jump over it.
- 131
4
votes
2
answers
420
views
Generating a group by randomly sampling generators
Let $G$ be a finite abelian group, $n$ a positive integer and let $G^n$ denote the direct product of $n$ copies of $G$. We say an element of $G^n$ is full if it acts as a nonidentity element of $G$ in ...
- 2,649
7
votes
2
answers
751
views
Looking for deterministic criteria to generate the symmetric group?
So let $S_N$ be the symmetric group of degree $N$. We think of it as a permutation group via its
natural action on the set $T=\{1,2,\ldots,N\}$.
Say that $H\leq S_N$ is a subgroup which acts ...
- 7,609