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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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About S3 generators relation in Artin's Algebra

I'm studying groups diving into the fantastic Artin's Algebra book. Actually, he's presenting the S3 group using the matrix representation of 2 permutations: $x = (1 3 2)$ and $y = (1 2)$ Actually, he correctly claims the whole group can be…
riccardoventrella
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Algorithm to write out all (anti-)cyclic permutations of a set

Is there any algorithm to systematically write out all the (anti-)cyclic permutations of, say the set $\{0,1,2,3\}$? I've already tried taking $(0,1,2,3)$, permuting it cycically, then transposing two of the numbers, say $(01)$, which yields…
Thomas Wening
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Uniqueness of sign function on permutations.

Let $f$ be any function which assigns to each permutation an integer number, and for any two permutations $\sigma$ and $\tau$, $f(\sigma \tau) = f(\sigma)f(\tau)$. Then, $f$ is zero, or identically 1, or it is the sign function. Solution After…
Majid
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Number of combinations for long AT&T password

AT&T gave me a ridiculously long, $11$ character password for wifi network. How many possibilities are there for this PW with $11$ digits? (Seems a super computer couldn't hack it in a hundred years). $2$ numbers, $2$ special characters, $6$ lower…
Robwest
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number of permutations in which $k$ numbers come after each other

I have $n$ numbers and I want to calculate the number of permutations in which $k$ certain numbers appear after each other (e.g. numbers $1$ to $10$ ($n = 10$), and let's say I want the permutations in which $6$ comes after $3$, and $3$ comes after…
RezAm
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Calculate $\alpha \sigma \alpha ^{-1}$

Exercise with permutations $$\alpha = (12)(135), \sigma = (1579)$$ Then $$\alpha \sigma \alpha ^{-1}= (12)(135)(1579)(531)(21)$$ In this case, I'm using permutation from right to left. I have the following steps $$1\to 2, 2\to2, 2\to2, 2\to2, 2\to…
Cure
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Permutation as disjoint cycles

I want to write the permutation $(123)(45)(16789)(15)$ as disjoint cycles. This is what I did: In standard notation, I believe it would be $$\sigma=\begin{pmatrix} 1 & 2 &3 & 4& 5&6 &7 &8 &9\\ 4&3&1&5&6&7&8&9&1 \end{pmatrix}$$ But how do I write…
Cure
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Find the number of ways the letters A,E,I,O,U,B,C,D,F,G can be arranged so that at least 4 vowels are together

Find the number of ways the letters A,E,I,O,U,B,C,D,F,G can be arranged so that at least 4 vowels are together Please assist in providing a step by step answer to help my daughter to understand the method to solve these kind of problems
Jay C
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Possible permutations, how to work out the maths

I am interested in understanding how to calculate possible permutations. For example for a 24bit MAC address (made up) ad:ba:32:d5:f0:dd. I believe the highest possible value per octet would be FF. So the highest possible value for the last 3 octets…
iNoob
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Permutation of α ∈ S9

Let $α ∈ S_9$ be the permutation given by $$α(1) = 6;α(2) = 8;α(3) = 7;α(4) = 9;$$ $$α(5) = 3;α(6) = 4;α(7) = 5;α(8) = 2;α(9) = 1.$$ Decompose the permutations α first as a product of disjoint cycles and then as a product of transpositions. Find…
soc
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How many arrangements of INCONSISTENT are there in which NE appear consecutively or NO appear consecutively, but not both NE and NO are consecutive?

How many arrangements of INCONSISTENT are there in which NE appear consecutively or NO appear consecutively but not both NE and NO are consecutive?
Y.Dan
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Ten people are to sit at a round table. Find the number of seating arrangements if the host and the hostess must always sit side by side.

The formula for circular permutation is $(n-1)!$, so if the two would sit next to each other, I'm not really sure of the computation. Would it be $(10-1)! \cdot 2!$ or $8!\cdot 2!$ ?
user530299
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Finding total possible permutations of n elements, in strings of n length (Math-Challenged person!)

Given the questions I see here, I'm guessing this will not be too difficult for this crowd - but, I'm math-challenged, so... All help much appreciated! A) How can I find the total number of permutations of a string of n length, when each element in…
CatharusGuttatus
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Permutations of a binary strings with even number of 1's

So I'm a bit rusty, but given a binary string, I'd like to calculate the permutations that exist which contain an even number of 1's within the range of binary 0 to said binary string. In other words, for all binary strings for the decimal range 0 -…
Jack Rohme
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How to distribute 8 workers on different weekends

A company has $8$ employees $p_1,p_2,p_3,p_4,p_5,p_6,p_7$ and $p_8$. Every weekend must be employed at least $3$ (or more) of them. These employees have the following restrictions: Employees $p_1,p_2$ and $p_3$ must work every other…
popi
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