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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Number of permutations $\alpha\in S_{n}$ with $\alpha^{2}=1$

This question is from Rotman's book An Introduction to the Theory of Groups. How many $\alpha \in S_{n}$ are there with $\alpha^{2}=1.$ (Hint. (ij)=(ji) and (ij)(kl)=(kl)(ij)) My try: 1) There is identity permutation. 2) Transpositions:…
user23505
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"Cyclic" and "Circular" permutation - Are they different concepts?

"Cyclic" and "Circular" permutations - are these two different concepts? I have been reading about permutation and encountering them in many places. What are the definitions of them, in simple English please. And what are the differences? Thanks in…
DotNet Developer
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Power of a permutation

Let $f = (1 2 3 4 5 6) ◦ (7 8 9 10) ∈ S_{10}$. Does there exist a positive integer n such that, when $f^n$ is decomposed into disjoint cycles, one of the cycles has length 5? Justify your answer. Any directions? Because I don't have the experience…
user600210
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Resolve into to factors

The number of ways the no. $10,800$ can be resolve as a product of $2$ factors, is My Try :: $10,800 = 2^4 \times 3^3 \times 5^2$ So Total no. of factors $ = (4+1)\times(3+1)\times(2+1) = 60$ But I did not understand how can i resolve into two…
juantheron
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Permutation count in cycle graphs

For $n \ge 3,$ Let $C_n$ be the cycle graph $(1,2,\cdots n)$ [where $n$ joins back to $1.$] Also let the distance $d(a,b)$ between two nodes $a,b$ be the lesser of $a-b$ mod $n$ and $b-a$ mod $n$. Thus for $n=10$ we have $d(1,10)=1,$ and also for…
coffeemath
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Permutation and Combination with divisibility?

How many five digit positive integers that are divisible by 3 can be formed using the digits 0, 1, 2, 3, 4 and 5, without any of the digits getting repeating? my explanation: total number of permutations with 0, 1, 2, 3, 4 and 5 to have 5 digits =…
munish
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Arranging a special permutation of the sequence (1,2,3..., n) into a circle

Given the first n natural numbers (1,2,3,...,n). We arrange a permutation of these numbers into a circle, such that the sum of any two adjacent numbers is prime. I have proven that there is at least one solution for n=4; n=6; n=8; n=10; I have tried…
ous Alam
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Permutations with conditions

I'm trying to solve a question involving permutations with conditions. I was reading the following problem: Five runners competed in a race: Fred, George, Hermione, Lavender, and Ron. Fred beat George. Hermione beat Lavender. Lavender beat…
vik1245
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Scheduling a Tournament

I am scheduling a games tournament for my girlfriend's cousin's 21st birtday party. There will be 8 teams. There can be either 7 or 8 different games. The briefing I have been given is: Each team must play every other team at least once; Each team…
Mark
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Find all possible permutations with duplicate items

I know how to find combinations of items where all items are different: $$P(n,r)=\frac{n!}{(n−r)!}$$ But how do I find it if some items are the same? For example, I have $3$ white and $3$ black elements and want to find out how many different…
Marked as Duplicate
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Permutations of Intersections of Parabola and Circles

What is the maximum number of points of intersection for four distinct parabolas and three distinct circles drawn on a sheet of paper? Anyone good enough to solve this.? Well considering the two different parabolas can intersect at 4 different…
D.Ronald
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Find the distinct number of arrangements of the symbols in the string###@@\$\$\$%%%% that begin and end with %

I have tried to solve this by excluding two % and performing permutation for non-distinct objects. As I excluded I left with 10 elements so n=10 and then considered non-distinct elements like # (n1=3),@(n2=2),$(n3=3),%(n4=2). n!/(n1!x n2!x n3!x…
R.Temur
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Longest cycle of permutation of $N$ elements

Given some permutation P of {1,2,3... N} we want to permute always by the same rule. What can be the largest K (number of permutation steps done) to obtain starting sequence? Eg. If we do 1->2, 2->3, 3->4 ... N->1, K will be N (as it is the number…
qwerty_
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Is this permutation even or odd?

Here is the question that I am working on: Let $\sigma$ be the permutation of the numbers $1,2,...,n$ which reverses their order completely. That is, $$\sigma=\begin{pmatrix} 1 & 2 & 3 &...&n \\ n & n-1 & n-2&...&1 \end{pmatrix}$$ Is $\sigma$…
John W. Smith
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How many ways can you rearrange the individuals in a row so that Soma and Eric don't sit next to each other?

Book: Probability For Dummies®, 2006, Rumsey, Deborah, PhD, Published by, Wiley Publishing, Inc., page 82 -- Extract from Google Books Problem: "Suppose you have four friends named Jim, Arun, Soma, and Eric. How many ways can you rearrange the…
mellow-yellow
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