-1
votes
0answers
73 views

Long cycles in $\text{Sym}(n)$ [on hold]

Let $x,y \in \text{Sym}(n)$ (symmetric group on $\{ 1,2,\ldots,n \}$) and $z:=xy$. Question: What non-trivial sufficient conditions (on $x$ and $y$) for $z$ to be a cycle of length $n$ do we know? ...
6
votes
1answer
226 views

Primitive, non-2-transitive groups with very large orbitals?

Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not ...
8
votes
1answer
215 views

Permutations of prescribed cycle types that multiply to the identity

Suppose that $\lambda_1,\lambda_2,\lambda_3$ are partititions of $n$. When do there exist permutations $\sigma_1,\sigma_2,\sigma_3 \in S_n$ such that (1) $\sigma_1\sigma_2\sigma_3$ is the identity; ...
5
votes
2answers
402 views

Permuting Racked Pool Balls with a Single Break

Given reasonable physical assumptions (on friction, collisions, etc.), would it be possible to "break" in a pool game such that when all the balls come to rest, the only difference is that the racked ...
6
votes
2answers
493 views

Two groups acting on a set.

Suppose we are given a set S of points on which two different groups G and G' (given by sets of generating permutations) act. Is there an efficient algorithm for finding generators the largest pair ...
5
votes
1answer
298 views

Counting the (additive) decompositions of a quadratic, symmetric, empty-diagonal and constant-line matrix into permutation matrices

Dear community, I have the following combinatorial question which I will explain in short first and then with some more detail. At the end you will find a very simple example. Short version Le $A ...