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

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3
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1answer
44 views

How to find a permutation of a specific rank?

I have a problem regarding permutations. When the rank of an unknown $S_7$ permutation is given, I want to find this permutation, but I can not. For example, I have the following questions: ...
0
votes
1answer
36 views

find the number of one-to-one function $[\pm n] \rightarrow [\pm n]$

the permutaion of $[\pm n]$ is a bijective (one-to-one) function $\pi:[\pm n] \rightarrow [\pm n]$ so that $\pi (-i) = -\pi(i)$ . $[\pm n]:=\{1, \dots, n-1, \dots, -n\}$. i have to find and determine ...
0
votes
2answers
47 views

Find a Four-element Abelian Subgroup of $S_5$ [duplicate]

Prof. Charles Pinter's "A Book of Abstract Algebra" provides this exercise: Ch 7 (Groups of Permutations) Part B #3 - Find a four-element abelian sub-group of $S_5$. Write its table. Please ...
4
votes
1answer
58 views

Which other “exotic” permutation-related things exist?

Some time back I posted some questions about the "exotic" outer automorphisms of $S_6$, and part of the answer was a citation of a paper by T. Y. Lam that said, among other things, that the ...
1
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4answers
99 views

How to write the set of all permutations on a set $n=\{1, 2, \ldots, n\}$

Let $n ∈ N$. Let $S_n$ denote the set of permutations on $\{1, . . . , n\}$. For any $σ ∈ S_n$, define $sign(σ) := (−1)^N$ , where $σ$ can be written as the product of $N$ transpositions. Now, let ...
6
votes
0answers
92 views

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$?

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$ so that each column and each row has exactly $n $ $1$'s and $n$ $-1$'s ? I tried for cases $n=1 , 2$ but the solutions were ...
1
vote
1answer
43 views

Number of ways to place 4 girls into 3 bedrooms.

A family has 4 girls and 3 bedrooms. 2 of the bedrooms are only big enough 1 girl, and the last room is big enough for 2 girls. How many ways are there to assign the girls to the bedrooms? I came up ...
9
votes
2answers
82 views

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 ...
2
votes
0answers
62 views

How many possible six-word sentences

A word is defined as a nonempty (possibly meaningless) sequence of letters. How many $6$-word sentences can be made using each of the $26$ letters of the alphabet exactly once? Generalise the result ...
4
votes
1answer
71 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...
0
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2answers
48 views

Trying to determine the number of possible combinations for a password

OVERVIEW: Making a secure password. People tend to use dictionary words as a basis for their passwords. People tend to make minor substitutions on their passwords (password -> p@$$w0rd) Assuming ...
2
votes
1answer
29 views

Combination or Permutation

Q.1)"Find the no. of ways in which $5$ boys and $3$ girls can be seated in a row so that no two girls are together." Q.2)"In how many ways can $5$ white balls and $3$ black balls are arranged in ...
2
votes
0answers
61 views

Characterization of conjugacy classes of $A_n$: intuition

Note the following theorem (quoted after handouts by Keith Conrad (UoCT) found online): Let $\pi \in A_n$. Its conjugacy class (cc) in $S_n$ remains the same in $A_n$, or it breaks into two cc's of ...
-2
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2answers
61 views

Combination and Permutations: How many ways can an award be given? [closed]

Have this Math question which I'm helping my cousin with but struggling to make sense of the answer. Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of ...
2
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0answers
46 views

Idemptent of Young Tableaux

I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. I understood that different Young Diagrams corresponds to inequivalent irreducible representations of ...
1
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1answer
39 views

Permutation with repetition and restriction

There are 5 red flowers, 4 blue flowers and 4 green ones. I must plant them so that no 2 red flowers are planted near each other. So I took all the possibilities (13!) and subtracted the ones where ...
1
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0answers
46 views

Permutation and Combination Problem-word arrangement

There are three pieces of paper.In the three papers ,a string (non-empty) has to be written such that none of the string on any paper is prefix of some other string.Also alphabet size of characters ...
1
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3answers
72 views

Number of different colourings of nodes

Consider a tree where each node has 2 subnodes, with a total of 7 nodes. So the maximum level of the tree is 2. Each node can be coloured white or black. Two colourings are equivalent if the one is ...
0
votes
1answer
36 views

A problem on permutation

Question :If there are $6$ periods in each working day of a school,in how many can one arrange $5$ subjects such that each subject is allowed at least one period? My solution: ${^5P_5} *{^5P_1}=600$ ...
0
votes
1answer
27 views

multiset/combination question

I have a bag full of: 7 green rocks, 12 yellow rocks, and 15 red rocks. How many ways are there to reach in and grab 4 rocks? Is the answer 37C34 (37=7+12+15+4-1) or 6C3 (6=3+4-1)...or something ...
4
votes
2answers
57 views

A question about irreducible representation of symmetric group (permutation group) in tensor space and tensor contraction

In chapter 13 of the textbook of Group Theory in Physics by Wu-Ki Tung, Lemma 2 discusses the equivalence of two irreducible representations of GL(m) on ${T^i}_j$. In its proof, it simply mentioned ...
0
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1answer
18 views

Number of permutations of a string with consequtive repetitions disallowed

Q: How many permutations of a string $AAABBCCDD$ exist such that consequtive characters $AAA$, $BB$, $CC$ and $DD$ don't appear in it. Note that $AA$ on its own is OK. A: The total number of ways in ...
0
votes
1answer
50 views

How to do permutation questions like this one:-

I am always confused on how to answer questions like this : Find the total number of possible permutations of all the letters of the word RESERVE. Find the number of these permutations in each of the ...
0
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0answers
28 views

Can there be two non-isomorphic sets of permutations with a one-to-one match between i's in S1 and k's in S2 (see description)?

In thinking about this question — where sets of $M$ permutations of length $N$ ($M<N$) are defined as "isomorphic" if one permutation function can be found that, when applied to each permutation in ...
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2answers
28 views

How many ways to come up with four teammates?

A Math teacher has to choose 4 people for a competition. If there are 6 boys and 6 girls, and the teacher must select 2 boys and 2 girls, how many ways are there? I came up with ${4 \choose 2} \cdot ...
1
vote
1answer
39 views

How many ways to line up if daughters are on sides of mother?

If we have a mother, father, 2 daughters and 3 sons lining up for a family photo, and the mother must be between the daughters, how many ways are there for the family to line up? I came up with ${5 ...
0
votes
2answers
119 views

how many ways can you divide 24 people into groups of two? [closed]

just can't seem to figure this out. I need to aquire a function for this scenario. I have tried to look at smaller forms of the problem. My problem is I am struggling to get the # of possibilities. ...
1
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2answers
35 views

Arrangements of sets of k positions in a n-competitors race

Let $E(n)$ be the set of all possible ending arrangements of a race of $n$ competitors. Obviously, because it's a race, each one of the $n$ competitors wants to win. Hence, the order of the ...
0
votes
0answers
11 views

What is the expectation of semi-fixed-points in a random permutation?

1<=i<=n is a semi-fixed point if: |π(i)-i| <= 1 with π of {1...n} What is the expectation of semi-fixed point?
1
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1answer
62 views

evaluating this expression with 3 variables .

I was solving a problem and I am stuck with this expression. As I am not a mathematics guy can someone please help me out with this : \begin{equation*} \sum^{n}_{i=1}\sum^{n}_{j=i}\sum^{n}_{k=j} ...
1
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2answers
86 views

How many ways to win this ternary row-game?

Sorry for the vague title. Please edit or comment if you know of a better one. Game description is below. I have a solution that works but coding it would be O($N!$) time complexity. I know there's a ...
-1
votes
1answer
18 views

$5$ variables, $6$ possible states of each, how do i calculate the number of possible states of this system [closed]

eg, $v_1$, $v_2$, $v_3$, $v_4$, $v_5$ can all take a value of $\{0,0.2,0.4,0.6,0.8,1\}$ independent of one another. How do I calculate or visualize the number of possible states of my system? ...
1
vote
1answer
27 views

Permutations, compositions and associativity properties

Let n be a postive integer, and let σ : {1, . . . , n} → {1, . . . , n} be a one-to-one and onto map. Then σ is called a permutation on n elements. The set of all permutations on n elements is denoted ...
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3answers
44 views

Is there any permutation $\tau\in S_7$ so that: $\tau^{4}=\sigma$?

Let $\tau$ be a permutation in $S_7$: $\ \sigma= \left( \begin{matrix} 1 & 2 & 3 & 4&5&6&7\\ 3 & 4 & 5 &6&1&7&2 \ \end{matrix} ...
3
votes
1answer
48 views

How many ways 5 different books be distributed among 5 students

I've seen this question in a book and can't figure it out correctly. Let 5 different books be distributed among 5 students. Suppose the books are returned and distributed to the students again ...
0
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0answers
24 views

Find a permutation $f\in G$ such that $h=fgf^{-1}$

I've been given two conjugate permutations $h,g \in G=S_{11}$ and have to find another permutation in $G$ such that $h=fgf^{-1}$. This seems similar to a change of basis for a matrix which I can do ...
5
votes
4answers
233 views

(combinatorics) prove that on average, n-permutations have Hn cycles without mathematical induction.

Prove that on average, n-permutations have $H_n$ cycles, where $H_n=1+1/2+1/3+...+1/n$ without mathematical induction. I think that on average, the number of cycles of length i (1≤i≤n) should be ...
1
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2answers
42 views

In how many ways can a $5 \times 5$ matrix be formed such that sum of row elements and column elements are $4$ and entries are $0$ or $1$?

Let we have a $5 \times 5$ matrix and the elements can be either $0$ or $1$ and the sum of elements of each row and column is $4$ then in how many ways can the matrix be formed ? I tried doing it in ...
0
votes
0answers
19 views

How to analyse the bound of the sum of permutation sequences?

suppose $X=[x_1, x_2, \ldots,x_n]$ ($0<x_1\leq x_2\leq \ldots\leq x_n$), and $$f(X) = \frac{x_1+2x_2+3x_3+\ldots+nx_n}{nx_1+(n-1)x_2+(n-2)x_3+\ldots+x_n}$$ i.e.,$$f(X) = ...
0
votes
0answers
55 views

finding number of subsets such that for given $(a,b)$ $a$ is the minimum element and $b$ is maximum element in that subset

I have a set of size $n$ which is sorted in ascending order. This is the process I followed: The largest element of the set is largest in $2^{n-1}$ subsets and the second largest is largest in ...
1
vote
1answer
29 views

Possible max matchings

our children (J/K/L/M) each wants a piece of fruit. There are five pieces of fruit available: an apple, a banana, a nectarine, an orange and a pear. J likes bananas and nectarines. K likes apples, ...
2
votes
1answer
32 views

Permutation At A Railway Track

Engines numbered 1, 2, ..., n are on the line at the left, and it is desired to rearrange(permute) the cars as they leave on the right-hand track. An engine that is on the spur track can be left ...
1
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1answer
37 views

Permutations how to eliminate with certain rules

I need to create a list with six elements $x$, $y$, $z$, $w$, $u$, $t$. After this, I should print all of the possible permutations of the elements with length $3$ which follows this rule: The ...
0
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1answer
46 views

Difficulty with a lemma needed to prove $A_n$ is a simple group for $n>4$

The theorem is: For $n \geq 5$, every normal subgroup $N$ of $A_n$ contains a $3$- cycle. The proof starts like this: Let $\sigma$ be an arbitrary element in a normal subgroup $N$. There are ...
1
vote
1answer
42 views

6 dogs and 4 cats enter a race, in how many ways can a dog finish first, second and third?

If using permutations 6*5*4 would give 120 ways that that dogs could occupy the first, second and third place. Is that correct?
1
vote
1answer
19 views

Order of a group calculation

Order of groups/permutations question, its very simple, but i'm having trouble understanding it. Why is the order of $(1372)(46)(5) : 4?$ By my understanding the LCM means its $4 \times 2 \times 1 ...
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0answers
12 views

Good resources for learning to recognize word problems in statistics?

I've got a number of books and resources for statistics theory, but I've always had problems with the approaches needed in answering questions, specifically for probability theory where counting ...
2
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4answers
46 views

Let $A= { x_1 , x_2 , x_3 , x_4 ,x_5 }$ , $B = { y_1 , y_2 , y_3 , y_4 , y_5 }$ , then find the number of one-one functions from $A$ to $B$ such that

Let $A= \{ x_1 , x_2 , x_3 , x_4 ,x_5 \}$ , $B = \{ y_1 , y_2 , y_3 , y_4 , y_5 \}$ , then find the number of one-one functions from $A$ to $B$ such that $f(x_i) \ne {y_i}$ where $i = 1,2,3,4,5$ . So ...
0
votes
2answers
30 views

Permutation and Combination 3 [closed]

Four different items have to be placed in three different boxes. In how many ways can it be done such that any box can have any number of items?
2
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1answer
41 views

Is it consistent without the axiom of choice that every permutation of some infinite set have fixed points?

A "permutation" of a non-empty set means an injective mapping of the set onto itself. Let $S(1)$ be the statement "There exists a permutation of every set containing at least two elements, which has ...