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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.

12854 questions
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Number of permutations on a lattice

Let's say I have a lattice like that: It's a lattice 10x10 (or N*N). So 100 little squares compose the lattice. Now I have to put 6 green squares (or n green squares) on the lattice. How do I calculate the number of possible permutations of the 6…
Rififi
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Permutations : If repetitions are allowed

For example if a question is to find the number of different ways of arranging $4$ letters of $26$-letter alphabet with repetition, I know that we have to do $26^4$. However, I am confused as to why exactly we are doing $26^4$. Are we assuming that…
Hamza750
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Writing permutations as products of disjoint cycles

How can I write these permutations as products of disjoint cycles? i.$\;\;(1234)(513)$ ii.$\;\;(13526)(53)(46215)$ iii.$\;(13)(12)(32)(143)$
orestiskim
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Permutation puzzler

Given a permutation on n letters, how many clues do you need to solve it? For example if the permutation is 31524, the clues come in the form of 5<4, meaning that 5 comes before 4. So, given a permutation $\sigma$ and a set of clues C: ($\sigma$,…
JMP
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Form of an element of a normal subgroup of $A_n$

I want to show that $A_n$ is simple for $n\geq 5$. For $n=5$ I have used the following criterion Let $H$ be a normal subgroup of $A_5$ then $H$ can contain any one of the following $a.$ a $5$ cycle of the form $(a,b,c,d,e)$ $b.$ a $3$ cycle of the…
Learnmore
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What is sgn(321)?

I've tried to compute the length of (321) and I got 2. Then the sgn should be (-1)^2=1. But I suppose sgn(321)=-1 by the definition in the graph?
Del Rey
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Finding cycles with set of permutations

Let $\alpha = (\alpha_1 \, \alpha_2 \, \ldots \, \alpha_s)$ be a cycle, for positive integers $\alpha_1 , \alpha_2 , \ldots , \alpha_s$. Let $\pi$ be any permutation. Show that $\pi \alpha \pi^{-1}$ is the cycle $(\pi(\alpha_1) \, \pi(\alpha_2) \,…
FlickS
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Example of an elementary permutation

Can someone please give an example of an elementary permutation? The book says that every permutation can be written as a composite of elementary permutations. Can someone please give an example? Thanks in advance!
Marion Crane
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Adding numbers in a consecutive series

I have the series: 1, 13, 133, 1333 ... Currently I have distributed it down to: 1 + (10 * 2) + (100 * 2) ... Can anyone point me in the right direction? Sorry, I forgot to mention, I'm looking for the sum of the series up to n where n is the stage.
fredthadinosaur
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combinatorial problem

What are the no of permutations for any no of adjacent elements swapping places at the same time in an array of length n?
Kaustav Dasgupta
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To calculate no of substring in length of 12 string

How many bit string of length 12 contains 01 as a substring ? I arrived at 2^10 . taking 01 as one set and remaining as other set
Techwatch
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Why is this $6!$ factorial and not $p(6,1)$?

There is this question. There are six different candidates for governor of a state. In how many different orders can the names if the candidates be printed on a ballot? The answer is $6!=720$. But why? If there are $6$ candidates and order matters,…
munchschair
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How many permutations given some condition(s)

Say you have six colored pencils: 2 green, 2 blue and 2 yellow and a given condition that no pencil of the same color can be next to each other How many ways to arrange the pencils if a. The pencils of the same color are identical. b. The…
Lightsout
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How many sequences of length N squared can be formed with N different values where each value is used exactly N times?

For instance, for N=2, the answer is 6 (e.g. aabb, abab abba baab baba bbaa). For N=3, the answer is 1680. I'm looking for the proper formula. Thanks
pkisensee
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Number of different possible permutations of a telephone number

A telephone number consists of $10$ digits, all from $0$ to $9$. The first digit is $0$. The remaining digits can be any number ranging from $0$ to $9$. How many possible telephone numbers are there? My try: I first said that since the first digit…
Aspiring Mathlete
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