Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [permutations]

For questions related to permutations, which can be viewed as re-ordering a collection of objects.

The word permutation has several possible meanings, based on context. In combinatorics, a permutation is generally taken to be a sequence containing every element from a finite set exactly once. Permutations of a finite set can be thought of as exactly the ways in which the elements of the set can be ordered.

In group theory, a permutation of a (not necessarily finite) set $S$ is a bijection $\sigma : S \to S$. The set of all permutations of $S$ forms a group under composition, called the symmetric group on $S$.

Reference: Permutation.

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Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence $(\hat{g}_{n,i})_{i=1}^n.$ For example, for $n =…
daniel
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Why are two permutations conjugate iff they have the same cycle structure?

I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?
Han Solo
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How many 7-note musical scales are possible within the 12-note system?

This combinatorial question has a musical motivation, which I provide below using as little musical jargon as I can. But first, I'll present a purely mathematical formulation for those not interested in the motivation: Define a signature as a…
MGA
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Can someone explain Cayley's Theorem step by step?

This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. Can someone explain this proof step by step?…
user72625
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$A_4$ has no subgroup of order $6$?

Can a kind algebraist offer an improvement to this sketch of a proof? Show that $A_4$ has no subgroup of order 6. Note, $|A_4|= 4!/2 =12$. Suppose $A_4>H, |H|=6$. Then $|A_4/H| = [A_4:H]=2$. So $H \vartriangleleft A_4$ so consider the…
Stephen Cox
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Odd/Even Permutations

How do you classify a permutation as odd or even (composition of an odd or even number of transpositions)? I somewhat understand the textbook definition of it but I'm having hard time conceptualizing and determining how it is actually determined if…
user65422
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Multiplication in Permutation Groups Written in Cyclic Notation

I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, if $$ a=(1\,3\,5\,2),\quad b=(2\,5\,6),\quad c=(1\,6\,3\,4), $$ then why does $ab=(1\,3\,5\,6)$ and…
com
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What is the shortest string that contains all permutations of an alphabet?

What is the shortest string $S$ over an alphabet of size $n$, such that every permutation of the alphabet is a substring of $S$? Edit 2019-10-17: This is called a superpermutation, and there is a wikipedia page that keeps track of the best results.…
Chao Xu
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$A_n$ is the only subgroup of $S_n$ of index $2$.

How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$? Why isn't there other possibility? Thanks :)
ShinyaSakai
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Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of…
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How does one compute the sign of a permutation?

The sign of a permutation $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula $${\rm sgn}(\sigma) = (-1)^m$$ where $m$ is the number of…
Chris Taylor
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The transpose of a permutation matrix is its inverse.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from assignment 4. Prove that the transpose of a…
jobrien929
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What is $\tau(A_n)$?

Suppose G is a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$. What is $\tau(A_n)$? Similar problems for some different classes of groups are already…
Chain Markov
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The pigeonhole principle and a professor who knows $9$ jokes and tells $3$ jokes per lecture

A professor knows $9$ jokes and tells $3$ jokes per lecture. Prove that in a course of $13$ lectures there is going to be a pair of jokes that will be told together in at least $2$ lectures. I've started with counting how many possibilities…
user565804
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Is a Sudoku a Cayley table for a group?

I want to know if the popular Sudoku puzzle is a Cayley table for a group. Methods I've looked at: Someone I've spoken to told me they're not because counting the number of puzzle solutions against the number of tables with certain permutations of…
Dan Glinski
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