All Questions
9 questions with no upvoted or accepted answers
27
votes
0
answers
940
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A question on simultaneous conjugation of permutations
Given $a,b\in S_n$ such that their commutator has at least $n-4$ fixed points, is there an element $z\in S_n$ such that $a^z=a^{-1}$, and $b^z=b^{-1}$? Here $a^z:=z^{-1}az$.
Magma says that the ...
- 271
23
votes
0
answers
1k
views
Do all possible trees arise as orbit trees of some permutation groups?
I.Motivation from descriptive set theory
(Contains some quotes from Maciej Malicki's paper.)
The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...
- 181
9
votes
0
answers
534
views
Generating $S_n$ with a fundamental transposition and a big cycle
I apologize in advance if this is too amateur, this is not really my area, but I'm very curious.
We have a permutation $\pi \in S_n$ and we want to represent it as a product of $\sigma = (1\;2)$ and $...
- 231
9
votes
0
answers
275
views
pattern-avoiding permutations vs multi-core partitions
Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Given the pattern $\sigma=k(k-1)\cdots321$, let $I_n(\sigma)$ be the number of involutions in $\mathfrak{S}_n$ that avoid the pattern $\sigma$. ...
- 43.2k
4
votes
0
answers
165
views
Counting "deflected" permutations: Part II
This is the second sequel to my earlier question on MO. Although the the current problem appears very similar, the answer is certainly different as experiments indicate.
As usual, let $\mathfrak{S}_n$...
- 43.2k
3
votes
0
answers
121
views
Twisted permutations
We consider a set $E$ with an involution (having perhaps fixed points).
We denote orbits by $\lbrace x,\overline{x}\rbrace$ (with $\overline{x}=x$ in
the case of a fixed point).
We consider sequences $...
- 17.9k
3
votes
0
answers
282
views
A new combinatorial problem for finite groups
In a recent preprint arXiv:1811.10503, I proved that if $a_1,\ldots,a_n$ are distinct elements of a torsion-free additive abelian group $G$, then there is a permutation $\pi\in S_n$ such that all ...
- 15.6k
2
votes
0
answers
85
views
Permutation factorizations according to number of generated orbits
Let $\pi$ be a permutation in $S_n$ with cycle type $\lambda$.
How many factorizations into two factors $\pi=\sigma_1\sigma_2$ are there, such that the subgroup $\langle \sigma_1,\sigma_2\rangle$ ...
- 2,552
0
votes
0
answers
126
views
Combinatorics of merging sequences from multinomial coefficients
If you have $m$ sequences $a_{11},\dots,a_{1n_1}$ through $a_{m1},\dots,a_{mn_m}$ each sorted in ascending order (assume there are no duplicates) then there is an unique way to merge them.
How many ...
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