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.

12854 questions
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What is the inverse cycle of permutation?

Given cyclic permutations, for example, $σ = (123)$, $σ_{2} = (45)$, what are the inverse cycles $σ^{-1}$, $σ_2^{-1}$? Regards.
JAN
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In how many of the integer numbers between 0 and 10,000 does the digit 3 appear to the left of 4

In how many of the integer numbers between $0$ and $10\,000$ does the digit $3$ appear some place to the left of the digit $4$? This would include, for example, the numbers $34$, $374$, $4384$ and $3874$, but would not include $27$, $43$, $3650$ or…
user322290
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2 answers

"Distance" between two permutations?

I've been trying to find a 'good' definition of the "distance" between two permutations that matches my intuition. I found this post which gets part of the way to what I'm thinking about, but I don't think it gets there entirely. I'm very open to…
user986122
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Exotic maps $S_5\to S_6$

This section says: There is a subgroup (indeed, $6$ conjugate subgroups) of $S_6$ which are abstractly isomorphic to $S_5$, At this point I'm thinking: certainly: the group of all permutations of $\{a,b,c,d,e,f\}$ that leave the letter $a$ fixed…
Michael Hardy
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every permutation is either even or odd,but not both

How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof... Thanks in advance...
DSC
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Integer solution $(x,y,z)$ to $xyz=24$

Find the total number of integer solutions $(x,y,z)\in\mathbb{Z}$ of the equation $xyz=24$. I have tried $xyz = 2^3 \cdot 3$ My Process: Factor $x$, $y$, and $z$ as $$ \begin{cases} x = 2^{x_1} \cdot 3^{y_1}\\ y = 2^{x_2} \cdot 3^{y_2}\\ z =…
juantheron
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11
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how many ways can the letters in ARRANGEMENT can be arranged

Using all the letters of the word ARRANGEMENT how many different words using all letters at a time can be made such that both A, both E, both R both N occur together .
darshanie M
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Distinct permutations of the word "toffee"

What does distinct permutations mean and how many distinct permutations can be formed from all the letters of word TOFFEE?
user1419170
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1 answer

Solutions of $ g p g = q $, with $p$, $g$, $q$ being permutations

Given two permutations of the same size, $ p $ and $ q $, what can be said about solutions to the equation $ g p g = q $ (with $ g $ being another permutation)? In particular, under what condition is there a solution? How many solutions are there?…
Command Master
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$(123)$ and $(132)$ are not in the same conjugacy class in $A_4$

Could you tell me how to show that $(123)$ and $(132)$ are not in the same conjugacy class in $A_4$? I know that all 3-cycles can't be in the same class, because the order of each class must divide $|A_4| = 12$ and there are eight 3-cycles in…
Bilbo
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I don't understand this permutation problem.

I'm trying to resolve this simple problem: We have $10$ beads, of which $8$ are white and $2$ are red and otherwise indistinguishable. In how many ways can we place the beads so that $2$ red beads are always separated? I think that this is a…
Mirko
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How many 4 digit number can be formed by 0,1,2,3,4,5 divisible by 4

How many 4 digit number can be formed by 0,1,2,3,4,5 divisible by 4 with repetition My Approach: Last two digits can be 00,04,12,20,24,32,40,44,52 that is 9 possibilities for last two digits. For the hundredth place digit all 6 possibilities…
justin takro
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8
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1 answer

Number of subsequences in a string

I know this might be one of the silliest questions out there but I'm going ahead and ask it here since I've lost practice in mathematics. I have been reading that the number of subsequences in a string is $2^x$ where $x$ is the length of the string…
lvella
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functions representable as a sum of two permutations

I am trying to prove that for every function $f:\mathbb{Z}/n\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ satisfying $\sum_if(i)=0$, there exist permutations $\pi_1, \pi_2:\mathbb{Z}/n\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ such that $f=\pi_1+\pi_2$. This is…
Martin Jeremies
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Calculating the power of permutations

I have this permutation $A$: $$ \left(\begin{array}{rrrrrrrrrr} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 10 & 8 & 5 & 2 & 3 & 1 & 6 & 4 & 7 & 9 \end{array}\right) $$ I want to calculate $A^9$. Is it ok to calculate it in this…
XXXX
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