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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Finding all results of a permutation group on a set

Given a finite group $G < Sym(\Omega)$; $\Omega$ finite, and $X \subset \Omega$, I can define a by the function $H(g) = \{x^g \| x \in X\}$ for each $g \in G$. Of course, each $H$ has the same size as $X$. Two different elements of $G$, $g_1,g_2$…
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In how many ways can an animal trainer arrange 5 lions and 4 tigers in a row so that no two lions are together?

Problem : In how many ways can an animal trainer arrange 5 lions and 4 tigers in a row so that no two lions are together? 1st Approach : L T L T L T L T L The 5 lions should be arranged in the 5 places marked 'L' This can be done in 5! ways. The…
Sachin
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Direct proof that Pr[2 immediately follows 1] in a random permutation is 1/n

The probability that $1$ is a fixed point of a random permutation of $\{1,2,\ldots,n\}$ (with uniform distribution) is $1/n.$ This is easy to prove since there are $(n-1)!$ permutations that have $1$ as a fixed point and $(n-1)!\,/\,n!=1/n.$ A…
user96124
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How to prove this assertion in $S_n$ for $n \geq 3$?

Let $n \geq 3$. Then there exists an element $f \in S_n$ such that $f \neq g^3$ for any element $g \in S_n$, where $S_n$ denotes the symmetric group on $n$ letters. How to establish whether this assertion is true or false?
Saaqib Mahmood
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Incorrect method to generate a random permutation of a finite set

The Wikipedia article on the Fisher Yates shuffle has a subsection on Potential Sources of Bias (Implementation Errors). It contains the following observation related to the possible outcomes when shuffling a three element array [1, 2, 3] using an…
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arrangement of NOT sitting together

I have the following exercise. Please help me to solve it. Exercise. In how many ways can 3 men and 3 women be seated at a round table if (a) no restriction is imposed (b) 2 particular women must not sit together (c) each woman is to be between 2…
user16168
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Is it possible to find sum of this series?

I am trying to find the sum of the following series asked by my…
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numbering permutations with repeated items

I want to number each permutation of ${1,1,1,1,0,0,0,0,0,0,0,0}$ (4 ones and 8 zeros) with a number. There are $\frac{12\cdot 11\cdot 10 \cdot 9}{4\cdot 3\cdot 2\cdot 1} = 495$ permutations. I want something like…
Jakube
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Prove that if $\sigma$ is the $m$-cycle $(a_{1}a_{2}...\, a_{m})$, then for all $i\in\{1,2,...m\}$ , $\sigma^{i}(a_{k})=a_{k+i}$

In course of my attempt of the following Dummit-Foote exercise of Symmetric Group: Prove that if $\sigma$ is the $m$-cycle $(a_{1}a_{2}...\, a_{m})$, then for all $i\in\{1,2,...m\}$ , $\sigma^{i}(a_{k})=a_{k+i}$, where $k+i$ is replaced by its…
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Find the number of ways in which 3 distinct numbers can be selected from the set $\{3^1,3^2,\dots,3^{100},3^{101}\}$ so that they form gp..

Find the number of ways in which $3$ distinct numbers can be selected from the set $\{3^1,3^2,\dots,3^{100},3^{101}\}$ so that they form gp.. My attempt. I tried to take numbers like $3^1,3^3,3^5$ (whose powers have a common difference of $2$ then…
maths lover
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Diagonal groups and semidirect products

I am studying a text on permutation groups, which has the following example in a section on regular normal subgroups: If $Z(N)=1$, then $N \cong \mathrm{Inn}(N)$, the group of inner automorphisms of $G$, and the semidirect product $N \rtimes…
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A group action contains another action as a normal subgroup

I have a question on what it means for one group action to contain another group action as a normal subgroup. I am guessing this means that the image of the first group action contains the image of the second as a normal subgroup, but I want to…
AG.
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Permutation/Combinations in bit Strings

I have a bit string with 10 letters, which can be {a, b, c}. How many bit strings can be made that have exactly 3 a's, or exactly 4 b's? I thought that it would be C(7,2) + C(6,2), but that's wrong (the answer is 24,600).
Andrew
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Show that H/K is isomorphic to $Z_2 \oplus Z_2$

We are given $H = \{(1),(13),(24),(12)(34),(13)(24),(14)(23),(1234),(1432)\}$ is a subgroup of $S_4$. Also assume $K = \{(1),(13)(24)\}$ is a normal subgroup of $H$. Show $H/K$ isomorphic to $Z_2\oplus Z_2$. This is just a practice question (not…
user77404
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How many 32 digit binary number combinations are possible?

How many 32 digit binary number combinations are possible? For…
Vishnu Vivek
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