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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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Transposition definition permutation

My book gives this definition. A permutation $z\in S_n$ is a transposition if: there exist $i,j\in[n]=\{1,2,3,\dots,n\}$ with $i\ne j$, $z(i)=j$ and $z(j)=i$ for all $k\in[n]$ with $k\ne i$ and $k\ne j$, $z(k)=k$ According to this definition, is…
deepinlife
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Difference in number of permutations between 3 substitutions and 4 substitutions

"A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. There are four substitutes: 1 goalkeeper, 1 defender, 1 midfielder and 1 forward. A substitute may only replace a player in the same category e.g. midfielder for…
Casper C.
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In how many ways can the word "permutations" be arranged if there are 4 letters between p and s?

In how many ways can the word "permutations" be arranged if there are 4 letters between p and s? I simply can't get any hold on this problem
Ayan Shah
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Show that $f\colon D_3\to S_3$ is an isomorphism

let $D_3$ be the dihedral group for a triangle (so it’s the set of all congruences that leave a regular triangle invariant). Consider $S_3$ the set of all permutations of 3 elements. Now my book says that it’s directly obvious that $f\colon D_3\to…
Sha Vuklia
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Permutation and Disjoint cycles question

I was given the following permutation \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 5 & 2 & 4 & 3 & 1 \\ \end{pmatrix} and I was asked to write it as the product of disjoint cycles. The disjoint cycles I found were $(1, 6)$ and $(2, 5,…
Smeef
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Round Table permutation with more chairs than people

A group of six boys and three girls went for a dinner after the show. They were given a round table for twelve people. a) Find the number of possible arrangements if seats are not numbered and the group can be seated without restriction. b)…
Rachana Murali Narayanan
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Permutations that mutual derangements of each other

Consider a sequence: $$S_0 = 1, 2, 3, ..N$$ Let $S_1$, $S_2$, $S_3$, ..., $S_k$ be $N$-length permutations of $S_0$ such that for any $(i,j)$ | $1 \le i < j \le k$, the sequences $S_i$ and $S_j$ are derangements of each other (i.e, they do not have…
Karthick
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How to calculate different permutations with multiple options?

I'm not a mathematician (although I do enjoy math). I'm a photographer. A wedding photographer to be exact. I'm currently re-vamping my album pricing and I want to show my clients just how customizable my albums are by giving them a number of…
Trevor Kennedy
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Number of permutation of natural number

Find the permutation of $\{1,2,3,4,5,6\}$ such that the pattern $13$ and $246$ do not appear. What I did in this question is found number of permutation of the $6$ given number I treated $13246$ as a single number and found the number of permutation…
user449276
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Calculating Permutations where n < r

I have a real life problem that sounds like word problem from an Algebra test, but I've haven't been able to find an example of how to solve it. My sister forgot the security code to the garage. The security code is 4 digits (order matters). She…
Grace B
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Writing the permutation as the product of disjoint cycles?

In my notes, it says that $\sigma = (1 6)(2 4 5 6)(1 4 5)(2 3)(7 8 9) = (1 3 2 5 6 4)(7 8 9)$. In my answer, I found that $1 \rightarrow 6$ then $6 \rightarrow 2$ then $2 \rightarrow 3$ and then 3 is fixed. So I got as far as (1 3 ...). Then I found…
mathstack
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Permutations differences - what's the name?

I'm solving some problem which is related to permutations and differences between adjacent elements in them. For permutation (1, 2, 3, 4) the differences I'm talking about would be (+1, +1, +1) For permutation (2, 3, 4, 1) the differences are (+1,…
Val Petruchek
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The sign of a composition of permutation

Let $f, g \in S_n$ I am to proof the following theorem: $$sgn(f\circ g) = sgn(f)\cdot sgn(g)$$ What I know is that $$sgn(f) = (-1)^{I(f)}$$ Where $I(f)$ is the number of transposition in permutation $f$. However I don't know how can the proof should…
Hendrra
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How many strings of length $4$ can be selected from $a, b, b, c, c, c, d, d, d, d, d$?

I came across problem I need some help with. Say I have this following 11 strings (with repeats) $a, b, b, c, c, c, d, d, d, d, d.$ Now the permutation is : $11!/(2! \times 3! \times 5!)$ that is something I know. But say I pick $4$ out of the $11$…
user1612986
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How many $5$-digit numbers can be formed from digits $0 ,1,....9$ such that no $2$ same digits are sit next to each other

How many $5$-digit numbers can be formed from digits $0 ,1,....9$ such that no $2$ same digits are sit next to each other? I tried to solve the problem but complement as following There are $$9 \cdot 10 \cdot 10 \cdot 10 \cdot 10$$ $5$-digit…
user373141
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