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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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Permutation Concept

There are five men and five women in the room. In how many ways can they be paired-up? How should I approach this question? The answer I have in my notes is 5!=120 ways which I don't quite understand. This is my thought process: 1. Choosing a pair…
Anlin yang
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Calculate probability some student has same number and same order

I have 10 students in the class, and have 18 numbers. Every student will get 9 random numbers from this set. $$\begin{array}{c|c|c} 7&10&1\\\hline 9&2&17\\\hline 8&5&7 \end{array}$$ If I'm not wrong, there are $9! = 362880$ possibilities the…
Anthony Dekoci
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Inversion count and number of transpositions

I'm given the permutation in $S_8$ $$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 7 & 8 & 4 & 1 & 2 &6 & 3 & 5\end{pmatrix}$$ and I've decomposed it into the transposition $$\sigma = (17)(73)(34)(28)(85)$$ and so $\sigma$ is an odd…
Twenty-six colours
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Equation with a permutation composition

Is there any method to solving such an equation: $$f_1\circ f = f_2$$ Where $f_1, f, f_2 \in S_7$ and: $f_1 = (1234)(5)(6)(7)$ $f_2 = (172536)(4)$
Hendrra
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Why is this the permutation written as the product of disjoint cycles?

In my notes, it says that $\sigma = (1 5)(2 4 7 5)(1 4 6)(2 3) = (1 3 2 6)(4 7 5)$ However the answer I get is $\sigma = (1 5)(2 4 7 5)(1 4 6)(2 3) = (1 4 6)(2 3)(7 5)$. What I did was I started with the smallest element, 1, and worked through the…
mathstack
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Find the nth permutation in dictionary order

What would be the most efficient way of solving the following problem? - A toy set contains blocks showing the numbers from 1 to 9. There are plenty of blocks showing each number and blocks showing the same number are indistinguishable. We want to…
7Aces
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Ways in which 16 people can be seated in a tea party

A tea party is arranged for 16 people along two sides of a long table with 8 chairs each side. four persons wish to sit on one particular and two on other side. in how many ways they all can be seated? now for four persons can sit on either side. so…
Taylor Ted
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The faces of an Icosahedron i.e; a regular polygon with 20 faces

The faces of an Icosahedron i.e; a regular polygon with 20 faces (each face is a equilateral triangle) are coloured with blue and white in such a way that no blue face is adjacent to another blue face. What is the maximum number of blue faces can…
MAS
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Verifying that the composition of permutations $(1, 2, 3) \circ (3, 4, 5) = (1, 2, 4, 5, 3)$

I want to make sure the following equality is true: $(1,2,3)\circ (3,4,5)= (1, 2, 4, 5, 3)$ I don't know how to check if that's true. Thank You
Sylvester Stallone
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Solve the product of these permutations.

This is just a simple product of permutations, can someone check my math? $$ (1234)(15792)(1932)(348)= (134)(2579)(8) $$ Yes? Where did I go wrong if it's incorrect? Here was my order of thinking(left to right):[1 to 9, 9 to 2, 2 to 3. Then 3 to 4,…
M.Matthews
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Disjoint Cycle Question

I'm working on this question: List the permutations of $\{1,2,3\}$ in disjoint cycle form. I already know what a disjoint cycle is. It's basically means that every cycle contains numbers that are not in any other cycle. So with that in mind, do I…
John W. Smith
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What meaning can be given to a permutation to the power of another permutation?

Say you have 2 permutations, f and g, how would one calculate $g^f$? Also can you multiply these permutations and how?
SFL
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Find the minimum.

If $x_1, x_2, ..., x_{100}$ and $y_1, y_2, ..., y_{100}$ are permutations of the set of numbers $1,2,3,...100$, what is the minimum value of $x_1y_1+x_2y_2+...x_{100}y_{100}$? I'm pretty sure the minimum is when $x_1, x_2, ..., x_{100} =…
user406996
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How can we prove that a cycle can't be produced by composing some cycles?

Let's think about a transposition $(3,\ 1).$ It can't be produced by composing $(3,\ 4,\ 1,\ 5),\ (3,\ 6,\ 1,\ 2),\ (6,\ 4,\ 2,\ 5),$ allowing zero or more use of them. How can we prove that?
Ris
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All permutation that commute with (1 4 5 2 ) (3 6) in S6?

How can i find all the permutations that commute with a given permutation. I know about the trick with multiplying by the inverse of the permutation but i don t know how to apply it? Can someone offer a step by step explanation?
Eduard6421
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