For questions related to permutations, which can be viewed as re-ordering a collection of objects.

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1answer
15 views

Conjugate permutation and “their” $\alpha$

Let $\sigma=(13624)(587)(9)$,$\tau =(15862)(394)(7)$. Determine such $\alpha$ that $\alpha \sigma \alpha^{-1} = \tau $. The elements $\sigma, \tau $ must be conjugate. But how many such $\alpha$ are ...
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0answers
7 views

Composition permutation of different cycle index

What can we say about the circle index of permutations resulting from the composition of two permutations of a different circle index?
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0answers
18 views

Average number of cycles in a uniformly selected random permutation of {1,…,n}

I (think) I'm on the right heading with this problem, but I feel like I'm taking a jump with my reasoning and relying on intuition. I've proved combinatorially that for a permutation of $\{1,...,n\}$ ...
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1answer
48 views

How to evaluate factorials greater than $69!$

How to evaluate factorials greater than $69!$? On my calculator, $69!$ is the largest number I can enter before it gives me a syntax error, most likely due to an overflow. Is there a way to evaluate ...
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0answers
15 views

Composition of permutations different types

What can we say about the type of permutations resulting from the composition of two permutations of a different type? Is the type of permutation must be the same as the type of reversed permutation? ...
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1answer
21 views

Prove that K is subgroup.

Let K be the set of all permutations of $ S_4 $ type $ [2 ^ 2] $ and the identity permutation $\in K $. Prove that $ K $ is a subgroup $ S_4 $. I would like to prove that for$\pi, p \in S_4$ and type ...
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0answers
9 views

Logarithm of an applied permutation

Say I have a cyclic permutation $P$, a known input $x$, and a known output $y$ such that $$y = P^a x$$ for some $a$. Is there a good way to search for $a$ (i.e. better than brute force)? Are some ...
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0answers
38 views

Permutations with forbidden values

I already asked this question in the mathoverflow forums, but it seems I won't get an answer as fast as I need one. So I'm moving the thread here. Besides, maybe it isn't as hard as to post it there. ...
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1answer
13 views

All possible types of permutation.

A permutation $ \sigma \in S_{10} $ satisfies the conditions $$ \forall_{1 \le i \le 29} \sigma^i \neq id, \sigma^{30} = id $$ Determine all possible types of the permutations. Give me a hand.
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0answers
25 views

Rectangular Seating Combinations [on hold]

Show me how I can seat 22 people in a rectangle with everyone sitting side by side. There are supposed to be 5 ways (different rectangles)? How do I do this?
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1answer
18 views

Finding permutation $a$ given $b$ and conjugate $a^b$

Normally we define a conjugate relationship as $$a^b = b~a~b^{-1}$$ But I don't know how to find $a$ given that we know $b$ and $a^b$.
1
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1answer
28 views

What is the minimal cardinality for a generating set of the permutations?

I want to find the minimum number of permutations so that all other permutations can be obtained by multiplying the permutations of this set (taken in any quantity). In other words, I am looking for ...
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1answer
11 views

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 ...
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0answers
13 views

conjugated permutations as to solve

We have: Niech $\sigma = (13624)(587)(9), \tau=(15862)(394)(7) $ Determine such permutation $\alpha$ that $ \alpha \sigma \alpha^{-1} = \tau$ How much are they?
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1answer
24 views

Order of a permutation divides n in Sn

Let $\theta \in S_n$, and for any $k \in \mathbb{N}$, either $\theta^k = I_{I(n)}$ or $\theta^k$ has no fixed elements. Show that $o(\theta) | n$. $I_{I(n)}$ denotes the identity. I'm completely ...
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1answer
26 views

Question in permutations

When we use this law? And in any case we use it? Thank you and I wish clarification.
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1answer
27 views

Permutations and their cycles [on hold]

What is the probability that k given elements belong to the same cycle in random permutation ? Thank you in advance.
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0answers
41 views

Prove probing to be permutation

So I have been taught that probe sequences (h(k, 0), h(k, 1), ... , h(k,m - 1)) are meant to be a permutation (0, 1, ... ,m - 1), but how does one prove that? I was asked this question in an ...
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2answers
31 views

Probability, why is this wrong? (Combinations and Permuations)

Why is this the wrong approach to solve this problem? "There are 65 students. 20 of them are sophomores, 20 are freshmen, 15 are juniors and 10 are seniors. When picking a 4 student committee, ...
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1answer
24 views

N Boxes and M babies question.

There are N boxes placed in a straight line. Adjacent boxes are separated by 1 unit. The Babies which are a total of M in number decide to play in this arena of boxes by moving from one box to ...
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2answers
21 views

In how many ways can a coat be chosen such that exactly one person picks up the correct coat?

There are n people at a party. At the end they each take a coat at random. a) How many ways can coats be chosen such that no person picks up their own coat/what is the probability that no person ...
1
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1answer
33 views

Find the number of elements of order 3 in $S_7$

I understand that there are two cycles of length 3, $(i,j,k)(a,b,c) \in$ $S_7$. However, I'm quite stumped in figuring out the logic behind these steps, leading to the answer : Number of distinct 3 ...
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1answer
20 views

In $S_6$, write the result as a product of disjoint cycles and then in the 2-row form.

(a) $(1,2,4)(4,3,5)(2,4)(1,2,4)^{-1}$ In the solution for this question, my professor has the product of disjoint cycles written as (1,3,5)(4,1). How would this make sense when disjoint cycles are ...
0
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1answer
16 views

permutations combinations

Q1. Total number of permutations of k diferent things , in a row , taken not more than r at a time(each thing may be repeated any no. of times) is equal to Q2. A teacher takes 3 children from her ...
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1answer
26 views

Show $\alpha^m = \varepsilon$ working with permutation groups

Show that $\alpha^m = \varepsilon$ using $\alpha^\ell (a_i) = a_{(i+\ell) \bmod{m}}$ where $\alpha = (a_0 a_1 \dots a_{m-1}) \in S_n$ a permutation group. I've been working on this problem but can't ...
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1answer
24 views

How to generate a single instance of multichoose (stars and bars)

So we know that if I have $k$ balls and $n$ buckets, I have $\binom{n+k-1}{k}$ unique ways to allocate the balls. Let's say $n=4$ and $k=2$ then I have $\binom{5}{2}=10$ ways. All possible allocations ...
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1answer
121 views

Pixel Permutations

How many possible arrangements of pixels can a 1024x768 pixel screen display if the color of a pixel is determined by mixing 3 values: red, green, and blue, ranging from an intensity of 0 to 255? The ...
2
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1answer
75 views

Number of possible permutations of n1 1's, n2 2's, n3 3's, n4 4's such that no two adjacent elements are same?

Given n1 number of 1's, n2 number of 2's, n3 number of 3's, n4 number of 4's. form a sequence using all these numbers such that two adjacent numbers should not be same. I have tries lot of things ...
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0answers
15 views

Permutation of conjoined faces in regular polygon with diagonals

I've been doing some study on relationships in polygons, right now, regular polygons. I've been trying to find relationships between the diagonals, angles, faces, vertices, and primarily conjoined ...
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3answers
84 views

Permutations and Combinations? 3 digit number…

1) Make a 3 digit even number without repeated digits, using 0, 4, 5 , 6, 7. Also the first digit cannot be 0. 2)Arrange 12 books in a line, 4 of which are english, 3 of which are science, and 5 ...
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1answer
73 views

Permutations and Combinations. Arranging things to be adjacent etc…

How many ways to do the following tasks 1)Arrange 12 blocks in a line, 4 of which are green, 3 of which are blue, and 5 red, so that all blocks are adjacent. 2)Form an 8 digit number using each ...
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0answers
15 views

Number of solutions for a multiple traveling salesman (mTSP) problem

Traveling salesman problem (TSP) with n-number of cities and only one salesman has "nPn" solutions which is n! but when you have more than one salesman, say k-number salesman, to travel n-number of ...
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1answer
14 views

Determine the isometric group $G$ which transfers a square into it self

I am solving the following exercise: Determine the isometric group $G$ of the euclidean plane which transfers a square into it self. The restriction of an element $g \in G$ on the vertices of ...
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3answers
28 views

Computing the inverse of a permutation

I didn't understand the permutation and of course, I got this question wrong. Compute the inverse of the following permutation: $$ \begin{pmatrix} 1&2&3&4&5&6\\ ...
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2answers
92 views

Permutaion and Combination Problems

Hey folks I'm having some issues with permutation and combination problems. 1) Make a 3 digit even number without repeated digits, using 0, 4, 5 , 6, 7. Also the first digit cannot be 0. I ...
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1answer
22 views

$5|\#\text{Gal}(f/\mathbb{Q})\subset S_5 \implies \text{Gal}(f/\mathbb{Q})$ contains a $5$-cycle?

Context: Consider $$ f(x):=x^5-4x+2\in\mathbb{Q}[x]. $$ By Eisenstein's criterion, $f$ is irreducible over $\mathbb{Q}$. Since $\mathbb{Q}$ has characteristic $0$, we know every irreducible polynomial ...
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2answers
43 views

Number of ways to group digits in {1,2,3,4,5,6,7,8,9} into numbers, while maintaining order

I have a set of integers from 1 to 9, call it A: $$A=[1,2,3,4,5,6,7,8,9]$$ How could I find the total number of possible combination of numbers within that set, while maintaining order? For example, ...
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1answer
26 views

Letter Arrangement with Permutations

In how many different ways can the letters of the word MAMMAL be rearranged so that the letters M are separated?
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1answer
34 views

How to find out if two cycles are conjugate?

Let for example $a=(14395)(26)(78)$ and $b = (154)(2368)(79)$ be elements of $S_9$. I know that by definition, conjugate elements of a group $G$ are elements $x,y \in G$ such that $x=aya^{-1}$ for ...
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3answers
38 views

$V_4\triangleleft S_4$

Let $V_4:=\{(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3),\iota\} \leq S_4$. It is possible to show $V_4\triangleleft S_4$ by considering conjugation. However, after long thought on the matter, I don't ...
0
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1answer
21 views

Painting parallelopiped with 6 different color

In how many ways can 6 faces of a rectangular parallelopiped with all 3 dimensions distinct , be painted with 6 different colours?? I have tried and i am getting 90 by $\displaystyle\frac{6!}{2^3}$. ...
2
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3answers
54 views

How many ways can the average of n dice be a?

For example, if n = 2, and a = 3.5 one could have (1,6), (2,5), (3,3), (4,3), (5, 2), (6,1) = 6 if n = 10 and a =3.5, one possible combination could be (1, 6, 1, 6, 1, 6, 1, 6, 1, 6)
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1answer
32 views

Partial Derangements

There are n people and n houses, such that every person owns exactly one distinct house. Out of these n people, k people are special (k<=n). You have to send every person to exactly one house such ...
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2answers
59 views

show elements of order 15 in $S_8$ satisfy a relation

Let $G$ be a group. Define a relation $\sim$ on $G$ by $a \sim b$ if there exists $g \in G$ such that $a = gbg^{-1}$. Prove that all elements of order 15 in $S_8$ are related by $\sim$. I noticed ...
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1answer
37 views

solving a simple problem of combination , with different approach

I found a question and I have different approach to solve it , but unable to get the answer. Question :How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible ...
3
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2answers
56 views

How many permutations of N integers with K pairs of numbers that can't be adjacent?

This is a computer science problem, I have a difficulty with the math part. There are $n$ integers $\{1, 2,\dots, n\}$ and $K$ pairs of numbers $(a, b)$; $a \ne b$; $a, b \le n$. No pairs are ...
0
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0answers
20 views

arranging three objects in six spaces

How many ways are there to arrange three objects in six spaces so no two objects are next to each other? I know the answer is 4 by doing it manually How can you tackle this problem using ...
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1answer
58 views

probability n! please help it's the only question i don't know how to do on the homework [closed]

write n! in terms of (n-1)! I am not sure what it is asking. I have tried everythng.
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1answer
16 views

Permutation of word

Question: Find the permutation of letters of the word EXERCISES in which vowels are together. My Efforts: I have rearranged the word in such a way that all the vowel come together. EEEI XRCSS Now ...