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

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Permutation of numbers that there are all modulo M .

Let's say I have $M-1$ integers, all of them different from each other, and all of them smaller than integer M: $$1,2,3...M-1$$ I multiply each of them by another integer S, and write the result ...
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1answer
36 views

How to use class equation for determining the center of $S_4$

How to use class equation for determining the center of $S_4$ $$|G|=|Z(G)|+\sum_x [G:C_G(x)]$$ So I guess I need to find $$|G|-\sum_x [G:C_G(x)]=|Z(G)|$$ Well $|S_4|=4!=24$ and $C_G(x)$ is the set ...
2
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1answer
28 views

transitive subgroups of the symmetric group on $2(d+1)$ elements: can I always do at least $d$ permutations?

I have a set of $2(d+1)$ elements which are labelled as pairs $\{e_i, a_i\}_{i=1}^{d+1}$, transforming under some transitive subgroup of the symmetric group. This can be thought of as a regular ...
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2answers
44 views

In how many ways can $5$ identical balls be placed in the cells of a $3 \times 3$ grid such that each row contains at least one ball?

In how many ways can $5$ identical balls be placed in the cells of a $3 \times 3$ grid such that each row contains at least 1 ball? I proceeded like this- In the first row choosing one cell out of ...
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2answers
35 views

Product of disjoint cycles and product of transpositions

$\alpha=(3412)(245)\in S_5$ and I have to 1) write this as a product of disjoint cycles, 2) write this as a product of transpositions. 1) I can do thing by following where the elements go in the two ...
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1answer
23 views

can i do this transformation with any finite group?

I have a finite alphabet $\{e_1, e_2, \cdot \cdot\cdot, e_N, a_1, a_2, \cdot\cdot\cdot, a_n \}$ where we pair $e_i$ and $a_i$ as ``opposites'' - like opposite vertexes on a regular $2N$ sided polygon ...
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1answer
22 views

Understanding representation of permutation matrix as vector

I hope this question is relevant here: I'm using some external software that does an LU decomposition of a square $(n\times n)$ matrix; the result is returned as three matrices L, U and P where P is ...
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1answer
27 views

permutations and probabilty

In a certain country, the number plate on a car consists of any 3 letters of the alphabet (the first letter is always a "K" or a "G"), followed by any 3 digits (0 to 9) and a alphabet. For a car ...
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2answers
22 views

Permutation with constrained repetititons

The question is as follows: How many ways can 12 identical white and 12 identical black pawns be placed on the black squares of an 8 x 8 chessboard My answer was $\frac{32!}{12!*12!}$ But the ...
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1answer
8 views

Rewrite permuatation as disjoint cycles

Rewrite $(3412)(245) \in S_4$ as a product of distinct cycles. I've only ever been given permutations as distinct cycles, transpositions or the matrix notation so I have no idea where to start.
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0answers
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Rook Polynomials with Symmetrical Overlap (Count Permutations Restricted by Distance)

Consider the cardinality $P(n,d)$ of permutations where elements can move up to distance $d$; for example, the permutation $\binom{012}{102}$ with $d = 1$ would be valid but $\binom{012}{201}$ would ...
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1answer
14 views

How to find all possible groups of four different values(integers)

I have four values :50,100,500,1000. I want to know many groups could be made with this combinations values. 50,100,500,1000 here it would be count as 1+1+1+1 50,50,50,100 count= 3+1 ...
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2answers
37 views

Normal subgroup in S4 [duplicate]

Let H be a subgroup of S4 where $H = \{e, B , C ,D \}$ $B(1)=2,B(2)=1,B(3)=4,B(4)=3$ $C(1)=3,C(2)=4,C(3)=1,C(4)=2$ $D(1)=4,D(2)=3,D(3)=2,D(4)=1$ Prove that H is a normal subgroup. I've tried ...
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2answers
62 views

How do I find the probability of some elements being together inside a randomly arranged set?

If I have a total of $n$ balls made of $k$ red balls and $(n-k)$ green balls and I arrange them all randomly in a line, how can I calculate the probability $x$ of a group of $y$ red balls being ...
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4answers
59 views

Sum of Digit Permutations

The question simply states "Let a secret three digit number be $cba$. If the sum of $cab + bac + bca + abc + acb = 2536$, what is $cba$?" I have no idea how to approach this problem. Any hints or help ...
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2answers
80 views

How many odd numbers less than $1000$ can be formed by using the digits $0,3,5,7$. Repetition not allowed. [closed]

Q. How many odd numbers less than $1000$ can be formed by using the digits $0,3,5,7$. Repetition not allowed. A. $21$ Answer is correct (please provide a thorough explanation). Unit digit nos. : ...
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3answers
40 views

We are given a class consisting of 4 boys and 4 girls.

We are given a class consisting of 4 boys and 4 girls. a committee that consists of a President, a Vice-President and a secretary is to be chosen among the 8 students of the class. Let a denote the ...
0
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1answer
38 views

The number of possible combination of column values with possibly common elements

I would like to calculate possible combinations for a given set of data: There is an x amount of columns (let's say 3) each column contains y amount of words (lets say 2), now I would like to ...
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2answers
57 views

Permutation of 6-digit numbers without repetition

How many 6-digit numbers without repetition of digits are there such that a ) the digits are all non-zero b ) 1 and 2 do not appear consecutively in either order ? Calculated the answer as below ...
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2answers
13 views

How do you find the order of a cyclic group?

What is the order of the cyclic group generated by $(1 2 5)(3 4)$? What is the order of the cyclic group generated by $(1 2 5)(3 5)$? I've looked through my notes and can't find notes on this and can ...
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0answers
17 views

Permutation in a grid without repetition

What is a general expression for the number of permutations of distinct objects, arranged in an m by n rectangular grid, such that no 2-element subset of a row is ever repeated from one permutation to ...
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3answers
211 views

Count permutations of $\{1,2,…,7\}$ without 4 consecutive numbers - is there a smart, elegant way to do this?

Here's a problem I've solved: Count permutations of $\{1,2,...,7\}$ without 4 consecutive numbers (e.g. 1,2,3,4). So I did it kinda brute-force way - let $A_i$ be the set of permutations of $[7]$, ...
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2answers
37 views

Permuation and Combination: Selecting n numbers such that such that sum is less than m

The value of n can be 0,1,2,3....and so on For example If we have to select 2 numbers such that the sum of all them can be less than 2 Manually the combinations can be (0,0), (0,1), (1,0), (2,0), ...
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2answers
54 views

Combinatorics of given alphabet

I'm looking for the formula to determine the number of possible words that can be formed with a fixed set of letters and some repeated letters. For instance take the 8-letter word SEASIDES and find ...
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3answers
28 views

Finding conjugacy classes

I've been having problems with finding conjugacy classes. I don't really understand how to do it properly. Say we look at a S3 group: $S_3=]e, (12), (13), (23), (123), (132)]$ If we look at just ...
2
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1answer
30 views

How to represent the sum of matrix elements given all permutations of a set of indices?

I would like to represent the sum all matrix elements of all permutations of indices given a set. For example, given the set $S=\{1,2,3\}$ I would like to compactly express ...
3
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3answers
320 views

How many four-digit odd numbers, all of digits different, can be formed from the digits 0 to 9, if there must be a 5 in the number?

How many four-digit odd numbers, all of digits different, can be formed from the digits 0 to 9, if there must be a 5 in the number? I know that there are 4 different cases where 5 is in the ...
2
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3answers
40 views

Cardinality of the set of automorphisms of $(\mathbb{N},+)$

I wonder if the set of bijections $\sigma\,:\mathbb{N}\to\mathbb{N}$ that satisfy $$ \sigma(a+b) = \sigma(a)+\sigma(b)\qquad \forall a,b\in\mathbb{N} $$ is countable or uncountable. What if we also ...
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2answers
29 views

Even permutations

I am given the symmetric group $S_{9}.$ Let $$\sigma = \begin{bmatrix} 1 & 2& 3& 4& 5& 6& 7&8 &9 \\ 4& 8& 7& 9& 3& 1& 2& 5 ...
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1answer
43 views

Composition of groups

Let's say we have a system of interacting particles that can divided into two populations. The symmetry group of each population is $G$, and the two populations are identical, so that I can exchange ...
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2answers
43 views

What is the best algorithm for finding a $g \in S_n$ which $a^g=b$ for given $a, b \in S_n$

What is the best algorithm for finding a $g \in S_n$ which $a^g=b$ for given $a, b \in S_n$, where $S_n$ is a symmetric group and $a$ and $b$ have same cycle type? Question 2: Is there any command in ...
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2answers
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Showing the permutation of 2 elements in a symmetry group is an even permutation

Show that for every 2 elements $\alpha$ and $\beta$ in $S_{8}$, the permutation $\alpha ^{-1}\beta ^{2}\alpha $ is an even permutation. How do I show that the above is an even permutation? I know ...
2
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1answer
50 views

Help with proof of the fact that $\det(A) = -\det(B)$, permutations

I got a couple questions regarding a proof of a well known property of the determinant. I'm not sure if the proof is correct (found it online): Proposition: If $B$ is a matrix gotten from $A$ by ...
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1answer
25 views

Help with understanding a proof on permutations

I've come across a theorem in Serge Lang's Linear Algebra, which I'm having trouble understanding. First I'll write the proof then indicate which part I do not understand: Proposition: Every ...
2
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1answer
34 views

square of a permutation cycle

$$\sigma = \begin{bmatrix} 1 &2 &3 &4 &5 &6 &7 &8 &9 \\ 1&5 &7 &4 &6 &9 &3 &2 &8 \end{bmatrix}$$ $$\sigma^{2} = ...
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3answers
137 views

Random Sequence of Alternating Increase/Decrease Numbers

The problem statement: Repeatedly pick a random number (uniformly-distributed) between $0$ and $1$. Keeping going while the second number is smaller than the first, the third number is larger than the ...
2
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2answers
35 views

Counting nearly-sorted permutations

Let $[n]$ denote the set $\{1,2,\ldots,n\}$. We call a permutation $\sigma:[n]\to[n]$, $(n,k$)-nearly sorted if $$\forall i\in [n]: |\sigma(i) - i|\le k,$$ i.e., every element is shifted at most ...
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0answers
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I need to prove that there is one homomorphism $\varphi : Dn \to Sn$ such that

I need to prove that there is one homomorphism $\varphi : Dn \to Sn$ such that $\varphi$($\tau$) = $$ \begin{pmatrix} 1 & 2 & . & .& . & n \\ 2 & 3 ...
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2answers
57 views

How many strings of $8$ English letters are there (repetition allowed)?

a) at least one vowel b) start with $x$ and at least one vowel c) start and end with $x$ and at least one vowel I can solve them easily by considering $total-no$ $vowel$. So, a) $26^8 -21^8$ b) ...
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1answer
35 views

How many strings of $3$ decimal digits have exactly two digits as $4$?

I approached the problem in this way : Fix two $4's$ and then third place can have $10$ ways to choose from {0,1,..,9} and then arrangement= ${10*3!}/2!$ = 30But , Since we have {$4,4,4$} and ...
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63 views

5 people into 8 seat train compartment

question for you all. In how many ways 5 people can be seated into an 8 seat train compartment, knowing that 2 people always sit by the window? would it be 6 choose 3 + 5 choose 2 ? that would give ...
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1answer
16 views

Question about Chapman's *An involution on derangements*

The (one-page-long) paper is available here: http://www.sciencedirect.com/science/article/pii/S0012365X00003101 To recap: For a permutation $\sigma$, we write $a_\sigma := \min \{ a \; | \; ...
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1answer
62 views

Hopefully simple permutations question

Say I have a permutation $B_{1}=(2,4,1,3).$ Now I'm thinking about a permutation $\sigma_{2}$ that gives me the permutation $(4,2,1,3)=\sigma_{1}$ from $B_{1}$, $\sigma_{2}B_{1}=\sigma_{1}.$ To ...
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3answers
57 views

Multiplying array elements

We are given a sorted array containing elements at indices $x_1,x_2,x_3,x_4,....x_n$. We have to find the product $\displaystyle\sum_{i,j,k}x_ix_jx_k$ where $j\geqslant i$ and $k\geqslant j$. For ...
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1answer
47 views

Professor has collection of $40$ issues of journal in $4$ boxes with $10$ issues per box.

Professor has collection of $40$ issues of journal in $4$ boxes with $10$ issues per box. How to distribute the journals if: $(a)$ each box is numbered $(b)$ boxes are identical I ...
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0answers
29 views

n positions to be filled by only x and y, such that no 2 x occur together [duplicate]

i'm not very good with permutations and combination, i need help with this problem. i need to use it in a programming question. Can someone derive a formula for "n positions to be filled by only x and ...
0
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1answer
50 views

An expression to represent all possible arrangments of a set of length n into 3 (infintely sized) bins?

I have a set, A = {1,2} And I generate a set, B, of all possible arrangements of the above set across 3 "bins" (note where 1 and 2 are together, they are summed): ...
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1answer
18 views

Need help understanding the precise meaning of “unique factorisation of disjoint cycles”

Below is taken from my linear algebra course lecture notes: Some facts about permutations of $\{1,2,\dots,n\}$: Every permutation is a product of disjoint cycles which commute. For example ...
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1answer
97 views

How many four-digit numbers can be arranged from the numbers {0, 1, 2, 3, 4}, when each number can be repeated a maximum of 3 times?

I just can't wrap my head around this. Maybe I'm overthinking it. I tried to use permutations with indistinguishable objects but I failed. Please help :(
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0answers
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Cycle Structure of a Permutation Based on the Binary Representation

Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number as follows. Given a non-negative integer $k$, let $s(k)=\frac{b+1}{2}$, where $b=\max\limits_c\big(c2^k\le n, ...