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

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Number of adjacent permutations.

What is the number of permutations for any number of adjacent elements swapping places at the same time in an array of length $n$? My solution: I think that we only need to count the number of ...
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1answer
25 views

permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$,

Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $G$. permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. ...
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2answers
21 views

Prove that there is a fixed point in any subgroup $H$ of $S_4$ of order $6$.

Prove that in every subgroup $H$ of $S_4$ of order 6 there is a fixed point in {$1,2,3,4$}, i.e, there exists $1\le i\le 4$ such that $h(i)=i$ $\forall h\in H$. $Start$: Suppose there is a subgroup ...
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1answer
36 views

In how many ways can the word “WORD” be rearranged so that no letter is in its original position?

In how many ways can the word "WORD" be rearranged so that no letter is in its original position? The answer is $9$, but what is the formula for it?
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Is there a name for these oscillations in the self-similarity of a set under the action of a cyclic group?

I don't know much about group theory and card-shuffling theory, so this may already have a name I don't know about. I often shuffle a deck of cards using a method that is defined by a particular ...
3
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1answer
44 views

Describe the subgroup $K\leq S_4$ of order 8

How do I construct the subgroup $K$ (a subgroup of $S_4$ of order $8$) ?
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0answers
28 views

Abelian minimal normal subgroup in a finite non-solvable group

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ an abelian minimal normal subgroup of $G$ ( $|N|=p^d$ for some integer $d$ and prime $p\neq 2,3,5$) such that $N=C_G(N)$, $Z(G)\leq N$ and ...
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1answer
30 views

cohomology of permutation group with mod 2 coefficient

Let $S_n$ be the permutation group of order $n$. Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. What is the cohomology algebra $$H^*(S_n;\mathbb{Z}_2)?$$ For $n=2$, $BS_2=\mathbb{R}P^\infty$ hence I ...
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0answers
25 views

Frobenius groups of order 36 [on hold]

Is there a Frobenius group of order 36? If yes, what is it's structure as semidirect product of two subgroups?
3
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1answer
37 views

Bringing a permutation back to the identity

I'm working with transposition distance (nothing to do with algebraic transpositions) on given permutations. Given a permutation, how many moves (transpositions) will it take to get back to the ...
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2answers
30 views

Number of elements of $S_{10}$ commuting with element (1 3 5 7 9) [duplicate]

Find Number of elements of $S_{10}$ commuting with element (1 3 5 7 9) I think we need to find order of centralizer of given permutation but how to find it?
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1answer
27 views

With $m>n$ , In how many ways $m$ men and $n$ women can seat in row for a photograph so that no two women are adjacent? [duplicate]

Given $m>n$ , In how many ways $ m$ men and $n$ women can seat in row for a photograph so that no two women are adjacent? My effort : There are $m-1$ gaps if $m$ men are seated. Now we have to ...
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1answer
34 views

Nth pemutation of Lexicographic String

Can someone please explain the logic behind the mathematical equation, that for finding the Nth Lexicographic rank of a string the Leading Entry is $a_q$ if $k=q\cdot (n!)+r.$ The link to the problem ...
2
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0answers
18 views

Sorting for maximum mean squared successive difference

I have a set of numbers and I have to order them for maximum MSSD (mean squared successive difference). For example, if I have the ordered set {1,2,3,4,5,6} this would give me an MSSD of ...
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1answer
35 views

Linear Permutations of $n$ objects

Suppose there are $n$ distinct objects $O_{1},O_{2},O_{3},\ldots,O_{n-1},O_{n}$. We have to find out the number of ways we can arrange them. But, there is a catch. We have to arrange them such that ...
1
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2answers
40 views

Number of circular permutation of word 'CIRCULAR' [closed]

Hey please help me with this question... Find the number of circular permutation of the word 'CIRCULAR'. Number of circular permutaion is (n-1)!
2
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2answers
40 views

Letters of the word “PARAMETER” [closed]

I have one question that bothers me. The total number of words that can be made by writing the letters of the word PARAMETER so that no vowel is between two consonants. The answer is 1800. I couldn't ...
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2answers
23 views

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
6
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1answer
51 views

Number of ways to pick N numbers from 0,1,…,N-1, with possible duplication, with sum equal 0 mod N

We have the numbers $0,1,2,....,N-1$ in $\mathbb Z_N.$ I want to pick $N$ numbers from these. These are the rules: Duplication may occur We don't care about ordering, $00041$ is equivalent to ...
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0answers
23 views

In how many ways can I move $M$ steps such that I do not leave the $N$-dimensional space at any point?

Suppose that currently I am at some position $(p_1,p_2,p_3 \dots p_N)$ in an $N$-dimensional space. The dimensions of the space is $(d_1,d_2, \dots d_N)$. In a step, I can walk one step ahead or ...
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0answers
47 views

Need help with simple probability question URGENT! [closed]

A confused frog has lost its way. Every 10 seconds it decides where to turn to. With probability 1/3 it jumps one metre to the right, with probability 1/3 it jumps one metre to to left, otherwise it ...
3
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1answer
104 views

Factorizing elements of a group into a product of generators.

$$ s = (1\ 2) \\ t = (1\ 2\ 3\ ...\ n) $$ Given the Symmetric Group $S_n$ generated by $s$ and $t$, is there a way to quickly factor an element $g \in S_n$ into a minimal product of positive powers ...
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2answers
14 views

Related question permuatation and combination 1

How to identify the question is permuatation or combination? And below is some question: i cannot solve. Show how to solve it. Thank you.
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1answer
22 views

probablity - number of const points in permutation

$\pi$ - random permutation. 1. Compute expected value of const points of $\pi$ 2. Compute Variety. 3. Estimate the probability that $\pi$ has more than $n/2$ const points Firstly, I have a problem ...
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1answer
39 views

Number of $4\times K$ board arrangements such that there exist no square of size 2 having all black cells

Suppose a person has a board of size 4 x K and each cell of this board can either be black or white. The person and his girlfriend come to a conclusion that they don’t like black squares. We decide to ...
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1answer
18 views

Permutation: How many numbers of n digits are possible for which product of its digits is a perfect square.

I need to find total numbers from 1 to 10000 whose product of digit is a perfect square. eg: 49 (4*9=36), 236 ( 2*3*6=36) etc. Till now i have figured out these things: 1) For a number to be a ...
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3answers
35 views

Is the map $f:S_n \to A_{n+2}$ given by $f(s)=s$ , when $s$ is even and $f(s)=s \circ (n+1 \space , \space n+2)$ , when $s$ is odd , a homomorphism?

Is the map $f:S_n \to A_{n+2}$ given by $f(s)=s$ , when $s$ is even and $f(s)=s \circ (n+1 \space , \space n+2)$ , when $s$ is odd an injective homomorphism ? I can show that if it is a homomorphism ...
2
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2answers
37 views

Problem on circular permutation

In how many ways can $x$ people be seated at a round table so that all will not have the same neighbours in any two arrangements?
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3answers
126 views

Seating permutations for 10 people where 2 people always sit together and 2 people never sit together

We have to seat 10 people in a row. Condition: two people always sit together and two people never sit together. My attempt: Let the two people who always sit together be taken as 1 person for the ...
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0answers
46 views

Number of permutations of order m

Is there a closed form for the number of permutations (on n letters) that have order m? If not, is there a tight upper bound?
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2answers
20 views

Permutations - n people and n seats

Actually, it seams pretty simple, but I just can't figure it out. Imagine we have a room containing $n$ seats in a row and $n$ people waiting in front of the room. The first person that enters the ...
0
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1answer
28 views

Florist bouqets of six flowers?

A florist is tying up bouquets of six flowers. The florist has ten white roses, ten orange roses, seven lillies, six tulips, four alstromerias and nine hyacinths at her disposal. How many different ...
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1answer
40 views

Discrete quotient group

I have a hard time understanding quotient groups. For example, I need to make sense of the expression $$\mathcal{S}_3 (1,3,5) / \mathcal{Z}_2 (3,5).$$ Here, $\mathcal{S}_3 (1,3,5)$ is a symmetric ...
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1answer
10 views

Dice game, number of permutations and overall probability

This question is on one of the past exam papers but I can't see to be able to get started on it. Here is the question: Alexi and Boris are playing a game of dice. Alexi rolls three dice and records ...
2
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1answer
27 views

Distinct vs Identical

In a bag containing 20 balls(6 red), (6 green), (8 purple) We draw 5 balls, put them back in the bag, then draw 5 more. In how many ways can this be done if the balls are considered distinct? My ...
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2answers
31 views

Number of permutations where an element must precede another

http://www.iarcs.org.in/inoi/2013/zio2013/zio2013-qpaper.pdf The fourth question. Can anyone explain how to solve it? I need to calculate the number of permutations possible while making sure that ...
2
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2answers
88 views

Arrangements of symbols with constraints

Symbols : $$1 , 2 , 3, 4 , a,b,c,d $$ Constraints :[$x<y$ means $x$ comes before $y$ in the arrangement] $1 < 2 < 3 < 4 $ $a < b < c < d$ $2 < c$ Find the number of ...
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1answer
25 views

Permutation nth term

I was working on a proof and I needed to know the nth term of the permutation's expansion meaning that: we all know that $^mP_n=m(m-1)(m-2)...(m-n+1)$ so if I want the $n^{th}$ term of that series, ...
2
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0answers
30 views

Maximum order of permutation of $n$ elements [duplicate]

What is the maximum order of a permutation of $n$ elements? The order of a permutation is the least common multiple of the lengths of its cycles, so this is the same as maximising LCM$(a_1,...,a_r)$ ...
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2answers
22 views

ordered pairs and combinations [closed]

The class has n students, n is an even number. The students are forming teams of 2. How many distinct ways are there to form the teams for the class? Write down the formula. Hint: when n=4, there are ...
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1answer
52 views

binary strings and counting sequence problem

Hello i'm working on these questions and I have few questions 1) A binary string is a finite sequence of 0 and 1. Ex. 001101 is a string of length 6 a) List all binary strings of length 4 (so I ...
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1answer
54 views

Representations of symmetric groups of order $2n$ and $n$

Background: Denote by $S_n$ the symmetric group of order $n$. There are many ways to embed $S_n$ as a subgroup into $S_{2n}$. Given a symmetric group, we can use Young diagrams to classify all ...
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2answers
51 views

making necklaces with beads problem

Hello im working on this problem and im completely stuck (a) there are 4 biz. How many necklase can I make? (b)consider the set of all necklase with distinct biz, where the size of a necklace is the ...
2
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1answer
61 views

different path permutation problem

Im having hard time with this question How many different paths in the $xy$-plane are there from $(0,0)$ to $(7,7)$ if a path proceeds either one space to the right $(r)$ or on space upward $(u)$? ...
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0answers
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Prove there is $\sigma\in S_3$ such that $H_{\sigma (i)} \cong\textrm{}K_i ,\space \forall i $

In class they gave us a problem, After spending a long time trying to solve it, I turn to you =] Let $H_1,H_2,H_3, K_1,K_2,K_3 \le G$ be simple groups, $G=\{{h_1}{h_2}{h_3}:h_1\in H_1,h_2\in ...
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1answer
36 views

Round table combinatorics?

Sorry if this is a horrible format to read(first time using this site!). Your friends A, B, C and D are going to sit right next to eachother around a round table at your birthdayparty. You do not ...
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0answers
50 views

Answer of a question

http://www.iarcs.org.in/inoi/2013/zio2013/zio2013-qpaper.pdf The fourth question. Can anyone explain how to solve it? EDIT: Sorry I didn't know about that. I need to calculate the number of ...
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1answer
18 views

Combination problem for N items in M identical groups

As the title says, I have N different items that will be put in M identical group, N >= M. The size of each group is not defined and the order of groups is not important. For example, if N = 4 and M ...
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2answers
50 views

sum of all the numbers that can be formed using the digits 2,3,3,4,4,4..

How to find the sum of all the numbers that can be formed using the digits 2,3,3,4,4,4.. What should be the way of doing this type of problem?? Please guide
0
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0answers
35 views

Given $k$ and $a$, find $n$ for $\frac{n!}{(n-k)!}=a$

Given the number of permutations of size $k$ for $n$ distinct objects $a$. Find the number of distinct objects $n$ (where $k \in \mathbb{Z}, 1 \leq k \leq n$) ? Basically, i'm given $k$ and $a$. I've ...