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For questions related to permutations, which can be viewed as re-ordering a collection of objects.
13,288 questions
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Structure of the permutations of certain type [closed]
Let $n\in \mathbb{N} $ be an even number and $\tau ,\rho\in S_n$ be two permutations such that $\tau$ is the product of $n/2$ disjoint cycles of length 2 and $\rho=(0,1,2,...,n-1)^t$ for some $t\in \...
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Action on left cosets by right multiplication
Let $G$ be a group and $H < G$ is a subgroup. If $\Gamma = \{gH \ | \ g \in G\}$ then $G$ act on $\Gamma$ by "left inverse" multiplication like $(gH) \circ t = t^{-1}gH$. But what happens ...
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Factorization pattern and cycle type.
Theorem. Let $p(x)\in\mathbb{Z}[x]$ have roots
$\alpha_1,\ldots,\alpha_n$ and $E$ as splitting field and let
$p^*(x)\in\mathbb{Z}_p[x]$ be the image of $p(x)$ under the natural
homomorphism $\phi_p$ ...
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1
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For which values of $n$ can the $n!$ permutations be arranged in a circle with matching adjacent letters?
Let $n ≥ 2$, and consider all n! permutations of $n$ distinct symbols (such as the letters $A, B, C,$ etc.). Suppose each permutation is written on a separate tile, and the tiles are to be arranged in ...
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1
answer
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Can all $𝑛 !$ permutations of $ n$ symbols be arranged in a circle with differing adjacent letters?
Suppose that for some collection of n distinct symbols A,B,C,…, each permutation of those symbols is written on a separate tile. Show that, for any n≥3, the resulting n! tiles can be arranged in a ...
- 153
2
votes
2
answers
87
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$9\times 9$ Go board symmetry- how can I think about this in terms of group theory?
I am learning about the game of Go- recently I started trying to learn $9\times 9$ go strategy.
According to the guide I'm learning from, there are only 15 unique starting moves on the $9\times 9$ ...
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1
answer
129
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Do all the transitive permutation groups of order $n$ arise as Cayley's embeddings of some group of order $n$?
Let $G$ be a finite group of order $n$. By Cayley's theorem, for any bijection $f\colon G\to X:=\{1,\dots,n\}$, $G$ embeds into $S_n$ as a transitive permutation group. Does the converse hold true? ...
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Permutation of first $2^n-1$ natural numbers and $\operatorname{lcm}(1,2,\dotsc,n)$
Let $\pi_n$ be a permutation of first $2^n-1$ natural numbers. Consider a step as permutting elements of permutation at the previous step as following: $\pi'_{n,2^{n-j-1}(2k+1)} = \pi_{n,2^j+k}$ where ...
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Permutation guessing game where you compare three indices at once
Here is a problem I encountered in the book "Olympiad Combinatorics".
Consider a game in which one person thinks of a permutation
of $\{1, 2,\ldots, n\}$ and the other person's task is to ...
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answers
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Determine visually if a finite group generated by cycles is cyclic. [closed]
Consider a subgroup $G$ of the permutation group $S_n$ generated by cycles $G=\langle c_1,\ldots, c_n\rangle$. I draw the points of the group as points in the plane and each cycle is drawn in a ...
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Arithmetic Progression in infinite permutations
I recently came across a question which asks to prove that each permutation of the set of all positive integers contains an increasing arithmetic progression of length three.
I was able to show so. ...
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1
vote
1
answer
70
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Simplified Social Golfer Problem ($4k$ golfers into groups of $4$)
I had someone talk to me about a problem which seems like a simplification of the social golfer problem, and I wondered if anybody here would have anything to add?
You have $4k$ people, and want to ...
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1
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1
answer
63
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How to find the maximum digit sum of the difference between two permutations of the same digits?
I recently came across an interesting property online that I hadn’t been aware of before: the difference between any two numbers formed by permuting the digits of the same number is always divisible ...
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Regarding the coefficients of characteristic polynomial of type D arrangements
We know in case of type A hyperplane arrangements the coefficients of the characteristic polynomial correspond directly to the Stirling numbers of the first kind. The Stirling number of the first kind ...
user982931
1
vote
1
answer
71
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Basics of permutation and combinations in 'genetics'
I am a biology student currently studying Mendelian genetics, but I am confused about how to use mathematical formulas to find various outcomes. This topic may seem basic, but my knowledge in this ...
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