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

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group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
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1answer
32 views

Product of $2$ permutations

$(2,3)(4,6,5,1,2)=?$ The multiplication is from right to left. I don't know, where I make the mistake. Denote $\tau=(2,3), \sigma=(4,6,5,1,2)$ $1\ \ 2\ \ 3\ \ 4\ \ 5\ \ 6$$\quad$ first apply ...
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1answer
30 views

Permutation inverse form

Given: $A=\{1,2,3,4,5,6\}$, $P_1=\begin{pmatrix} 1 &2& 3& 4& 5& 6\\ 2& 3& 4& 1& 5& 6\end{pmatrix}$, $P_2=\begin{pmatrix}1 &2 &3 &4& 5 ...
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2answers
86 views

Solving Rubik's cube and other permutation puzzles

I've seen two questions on solving the Rubik's cube but none of the answers have given a complete solution using mainly mathematical techniques. Furthermore, I've not seen a good explanation of ...
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1answer
57 views

Subgroups of $S_n$ that can send any subset of $[n]$ to any equally sized subset of $[n]$

This is a repost of a question I was trying to solve yesterday that got deleted. The question asked for a characterization of the subgroups $G$ of $S_n$ which when endowed with their natural action on ...
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3answers
35 views

Permutation in discrete math

Is the permutation $$\begin{pmatrix} 1& 2 &3 &4 &5 &6&7 \\ 7 & 4 & 2 & 1 & 3 & 6 & 5 \end{pmatrix}$$ even or odd? The product of disjoint cycles is ...
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5answers
784 views

How many Arrangement of “AMAZED” letter E Positioned between two A's (Not necessarily Flanked)

I considered 'AEA' as one letter so there are 4 letters which can be arranged in 4!=24 ways. But my sheet is telling its 120 How? Please HELP! & What is flanked meaning here?
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1answer
20 views

Maximal Permutations of numbers with monotonic objective function

I felt confident in the validity of the following statement, but now that I've played with the proof more I'm starting to have a few minor doubts. Any thoughts? Suppose you have two partitions of ...
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1answer
31 views

Write a transposition as a product of adjacent transpositions

I just read that any transposition can be written as a product of adjacent transpositions. I thought that I knew the right proof of this, but then I read that $\tau_{i,j} = \tau_{i,i-1} \circ...\circ ...
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0answers
34 views

Circular arrangement problem

I have one question of circular round table arrangement: " How to find the number of ways in which 6 persons out of 5 men and 5 women can be seated at around table such that 2 men are never together. ...
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1answer
29 views

Derangement, example, paradox?

How to explain that $!0 =1 $ and $!1=0$ I understand why $0! =1$. But when it comes to permutation $1!$ is also $1$. And in result it doesn't argue with my intuition. But, when it comes to ...
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1answer
86 views

When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated? [3] [duplicate]

Let $A$ be an $n\times n$ matrix with entries in the set $\{0, 1\}$ which has exactly two ones in each column and in each row. Give necessary and sufficient conditions for the rank of $A$ to be $n$.
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2answers
67 views

Please explain definition of determinant using permutations?

Many people (in different texts) use the following famous definition of the determinant of a matrix $A$: \begin{align*} \det(A) = \sum_{\tau \in ...
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1answer
25 views

Permutations and combinations - number of ways to pay

Question: 22 people go to a movie theater. 11 of them are carrying a 50 dollar bill while the other 11 are carrying a 100 dollar bill. The ticket for the movie theater costs $50. The cashier ...
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1answer
56 views

Seat 5 men and 4 women in a row such that the women occupy even places [closed]

It is required to seat 5 men and 4 women in a row such that women occupy even places. How many such arrangements are possible?
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3answers
98 views

Textbooks on permutation groups?

I need good texts on group theory that cover the theory of permutation groups. I think there is a book called Wielandt. Is it good? are there newer alternatives? Can I find books that are not ...
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2answers
50 views

Confusion in combinatorics

Question (1) The number of different ways in which $10$ telegrams can be distributed to 2 message boys is ____? The answer as per the book is $2^{10}$. But, I think answer should be $10^{2}$. If ...
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2answers
35 views

How many ways to divide 11 people into 3 groups of 3 and one group of 2? [closed]

How many ways are there to divide 11 people into 3 groups of 3 and one group of 2? The right answer is 15400 but I can't get it
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1answer
29 views

Permutations and combinations - choosing an integer

Question: In how many ways can we choose 2 distinct integers from 1 to 100 such that the difference between them is at most 10? Approach: I tried to fix a certain number, and then find the ...
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1answer
34 views

What's the sign of $\det\phi'=\pm 1$ where $\phi:\mathbb{R}^n\to\mathbb{R}^n$ is a permutation of coordinates?

Let $S_n$ denote the symmetric group and $$\phi:\mathbb{R}^n\to\mathbb{R}^n\;,\;\;\;x\mapsto\left(\begin{array}{c}x_{\pi(1)}\\\vdots\\x_{\pi(n)}\end{array}\right)$$ for some $\pi\in S_n$. Obviously, ...
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1answer
374 views

Count permutations with LCM

Given $N,M$ and $D$ we need to count how many permutations of $N$ integers are there with each $i$'th element $1 \le A[i] \le M$ such that least common multiple (LCM) of all its elements is divisible ...
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0answers
24 views

Permutations and combinations - divisibility of a factorial

Question: Find the largest value of $n$ for which $125!$ is divisible by $6^n$ Approach: I tried to find all the numbers which were a multiple of 6. The number of times such divisors occurred ...
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2answers
26 views

Permutations and combinations - number of solutions

Question: Find the number of solutions of $x_1+x_2+x_3 = 51$ for $x_1,x_2,x_3$ being odd numbers Not sure how I would even begin this question. It would be simple except for the condition given ...
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1answer
29 views

Normalizer of a subgroup generated by a cycle.

Let $H$ be the cyclic subgroup of $S_4$ generated by the cycle $(1234)$. What is the order of the normalizer $N$ of $H$ in $S_4$? Give generators for $N$. How do I go about solving a ...
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2answers
148 views

Permutations and combinations - distributing objects into groups

Question: In how many ways can $2n$ people be divided into $n$ pairs? Approach: Well as there are $2n$ people, it is obvious that we need to chose each and everyone of them. Using simple ...
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0answers
48 views

25 books - permutation question

Imagine 25 books, 5 groups of books (e.g. maths, biology, history, geography, philosophy...) and all groups has 5 seperate colors of books (black, blue, yellow, red, green). So there are 5 blue books ...
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1answer
37 views

Are the primitive groups linearly primitive?

A transitive permutation group $G \subset S_n$ is primitive if $G_1 \subset G$ is a maximal subgroup. A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
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0answers
78 views

Is there a transitive permutation group satisfying these properties?

Let $G \subset S_n$ be a transitive permutation group and let $H=G_1:=\{ g \in G \ \vert \ g(1)=1 \}$. Let $(K_i)_{i \in I}$ be the sequence of minimal overgroups of $H$ in $G$. Note that if $G$ is ...
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2answers
41 views

Number of zeroes at end of factorial

Question: How many zeroes will there be at the end of $(127)!$ Approach: Considering the fact that when two numbers ending in $x$ and $y$ zeroes are multiplied, the resulting number contains $x+y$ ...
2
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1answer
31 views

How many combinations can a group of n people form?

how many groups can $20$ (or $n$) people form? The size of the groups vary from $2$ to $20$, where no group should have only a single member, and the order doesn't matter I'm sorry if this is a ...
3
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0answers
77 views

Number of $n$-permutations for which ${\tau}^k = id$

I am curious about the formula(any closed form) for the number of $n$-permutations $\tau$ such that ${\tau}^{n-1} = id$. How about for the case ${\tau}^n = id$ ?
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0answers
47 views

Confusion in a Mathematics Symbol

I fail to understand what a certain symbol means, please explain. It is in my permutation and combination chapter and it is for arrangements of 7 digits where something repeats thrice and something ...
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0answers
21 views

hy $\langle g\rangle$ is $p$-sylow subgroup of $G_\Delta$?

Let $p$ be a prime and $G$ a primitive group of degree $n=p+k$ with $k\geq3$. If $G$ contains an element of degree and order $p$. $G$ contains the cycle $(1,2 ... p)=g$. Let $\Delta= \lbrace ...
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0answers
16 views

unique children of a point in a boolean lattice

I am working with two-element boolean algebra, e.g. points composed of strings of $0$s and $1$s and bit-wise $AND$ and $OR$ to find maxima and minima. In the domain I'm working in, I need to assign ...
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2answers
142 views

Commuting permutations

I saw the following statement in a book: If $\sigma, \tau\in S_n$ such that $\sigma\tau=\tau\sigma$, then the order of $\phi=\sigma\tau$ is the least common multiple of the orders of the permutations ...
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0answers
37 views

Conjugacy class A(4)

I want to find all conjugacy classes of $A(4)$. So basically what I did, I took all elements of $A(4)$ and calculated their conjugates. I had no problems with $$\{e\}, \{(123),(134),(142),(243)\}, ...
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1answer
59 views

general formula for permutations with some ordering

Assume I want all permutations of a set of numbers with certain numbers must go before others. Similar to this question but I'm looking for a more general formula. For example the set ...
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0answers
41 views

Is my solution correct? (If $S_\Omega \cong S_\Delta$ then $|\Omega|=|\Delta|$)

I've tried to solve the following exercise from Dummit & Foote's Abstract Algebra text (p. 151, Exercise 8). Here $\Omega$ is an infinite set, $D \leq S_\Omega$ is the subgroup of permutations ...
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0answers
23 views

we can say $G_\Delta$ is primitiveon $\Omega-\Delta$?

Let p be a prime and G a primitive group of degree n=p+k with k≥3. if G contains an element of degree and order p، then we can say $G_\Delta$ is primitiveon $\Omega-\Delta$ ? ...
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0answers
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proof theorem 13.9 on finite permutation groups of Wielandt book

I can't prove this theorem on finite permutation groups of Wielandt book. Theorem 13.9: Let p be a prime and G a primitive group of degree n=p+k with k≥3. if G contains an element of degree and order ...
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2answers
29 views

Permutations and Combinations Doubt

Question: How many words, with or without meaning, can be made using the letters of the word DEBOTRI such that there are always two letters between D and E? I got $4 \times 2 \times 5P_5 = 960$, ...
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0answers
26 views

Number of words that can be formed using the word PHILOSOPHY [duplicate]

In the word PHILOSOPHY how many words would have the letters H,I,S,Y together when words are formed by using all 10 letters? ...
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1answer
26 views

number of strings formed of k characters and length n

You have been given a set of characters of size $k$ and you have to make strings of length $n$. How many strings are possible. A constraint is that every character must be used at least once in each ...
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2answers
59 views

Sum of binomial coefficients $\sum _{ x=r-2 }^{ n-2 } \binom{x}{r-2}$

$$\sum _{ x=r-2 }^{ n-2 } \binom{x}{r-2}$$ I can't find the sum of the following series. I would appreciate if anyone can show me this problem's solution.
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1answer
23 views

Are bit permutations linear?

I'm trying to answer a question on cryptography. Basically, inputs and outputs (so plaintexts and ciphertexts) are from the set (say) $\{0, 1\}^{100}$. The encryption function takes inputs and splits ...
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2answers
30 views

Permutations and Combinations - divisors of a number

Question: A number $n$ is given as $2^{31}3^{19}$. Find the number of divisors of $n^2$ which are less than $n$ and not a divisor of $n$. Well the total number of divisors of a number is given by ...
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4answers
76 views

Number of 6 digit numbers with digits 1,2,3,4 with each digit appearing at least once.

Find the number of 6 digit numbers that can be made with the digits 1,2,3,4 if all the digits are to appear in the number at least once. This is what I did - I fixed four of the digits to be 1,2,3,4 ...
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1answer
58 views

Xmas Special: 25 identical sweets shared between two indivuduals and…

Ready?.. 25 identical sweets must be shared bewteen 1 boy & 1 girl. Each of the children MUST recieve at least 10 sweets each. all sweets must be distrubuted. order not important although I will ...
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1answer
31 views

Algorithm for unbiased random derangement

I am looking for an algorithm that generates a derangement with uniform probability across all possible derangements. This is similar to Generating a random derangement, but with the requirement that ...
2
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1answer
81 views

On a proof that “there are at least $F_n$ Collatz permutations of length $n$”.

Let $n, k \in \Bbb{N}$ and $F_n$ be the $n$th term of the Fibonacci sequence. Let $u$ be the map $x \to 3x+1$ and $d$ be the map $x \to \frac{x}{2}$. Let a type be a sequence of $u$'s and $d$'s. ...