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

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15 views

Number of ways of selecting a submatrix?

Given a matrix $N \times M$ and a point $(x,y)$ then in how many ways can you select a submatrix such that the point lies inside the submatrix ?
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2answers
19 views

Permutations with repeated item [on hold]

I have: Orange Apple Orange Guava Pineapple Watermelon Strawberry Note the repeated Orange. Out of these 7, I have to choose 4 fruits. How many permutations are possible?
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1answer
17 views

Compute the matrix with the standard basis.

Let $X$ be the left representation of $S_3$. Compute the matrix $X(132)$ in the standard basis $S = {\{123,132,213,231,312,321}\}$. attempt: In general if $G = S_3 = {\{\pi_1, \pi_2,..,\pi_6}\}$, ...
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2answers
21 views

Multiplication of transpositions?

I can't seem to understand how the multiplication of two transpositions yield the results below: $(x b)(x a) = (x a)(a b) \\ (c a)(x a) = (x c)(c a)$ I can't figure it out for the life of me. I'm ...
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0answers
41 views

Proof involving the group of permutations of $\{1,2,3,4\}$.

Let $\sigma_4$ denote the group of permutations of $\{1,2,3,4\}$ and consider the following elements in $\sigma_4$: ...
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1answer
22 views

What does it mean a transitive permutation?

let $X$ be a finite set. Let G be a group. What is the meaning of $G$ is a transitive permutation on set $X$?
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2answers
26 views

What is the probability that nobody receives the same ranking twice?

Four players compete in a tournament and are ranked $1$ to $4$. They then compete in another tournament and are again ranked from $1$ to $4$. Suppose that their performances in the second ...
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1answer
32 views

Order of a subgroup generated by two permutations

Let $$\alpha=(1,3,12,7)(8,5,6,2,11)(4,9,10)$$ $$\beta = (1,5)(6,8,11)(12,3,2,7)(4,9,10).$$ How does one prove that a subgroup G that contains $\alpha$ and $\beta$ has order $o \ge 120$? ...
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34 views

Permutations of $S_n$ whose order divides a positive integer $m$

For which $n,m \in \mathbb{N}$ is $$K_m = \{\sigma \in S_{n}: \text{ord}(\sigma) \text{ divides } m \}$$ a subgroup of $S_{n}$? How does one approach such a problem?
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0answers
37 views

Hausdorff quotient space

consider a smooth manifold $M$ and a group action, i.e. a group homomorphism $\phi: G\rightarrow S(M)$, where $S(M)$ denotes the group of diffeomorphisms of $M$. Suppose that for all $K\subset M$ ...
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22 views

License Plate Permutations

A state has changed its license plate numbering system for the three largest counties. Before the change, each plate had the number 1, 2, or 3, followed by either one or two letters, followed by 3 ...
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2answers
39 views

What is the probability that when a deck of cards is shuffled and dealt, exactly 3 of the 4 aces will be dealt within the last 20 cards? [on hold]

I am trying to figure out this problem, I think that it is a "permutations with repetition" type of question.
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16 views

Disjoint cycle Decomposition.

Let $\sigma =(a_1...a_6)$ be a $6$-cycle. Write the disjoint cycle decomposition of $\sigma^2$ and $\sigma^3$. I know that $a_1*a_1=\epsilon $ but does this mean that $\sigma^2=\epsilon$ and then ...
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0answers
28 views

Looking for mathematical/combinatorial and computational explanation regarding adding values in a $5 \times 4$ (matrix?) with a constraint.

Given the following matrix (not sure if I should call it that): Matrix $5 \times 4$ I want to add all possible combinations of values such that each Horse gets but one value from each Bookie. What I ...
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2answers
20 views

Why use C(n,r) instead of P(n,r) when considering how many strings can be formed in which a specific letter appears before another specific letter?

I am dealing with a problem in which I must determine how many strings can be formed by ordering the letters ABCDE subject to the conditions given. The condition that I am given is that A appears ...
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20 views

permutations and representations , sign function.

Show that the sign representation of $S_n$ is indeed a representation. attempt: Recall the sign function of a permutation is given by $\mathrm{sgn}(\pi) = (-1)^k$. Then recall a representation is a ...
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73 views

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? [on hold]

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? How should I approach this problem? Okay I think it's C(10, 2) because I already have ...
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0answers
24 views

Centralizer of $(1,2,3)(4,5,6,7,8) \in A_{11}$

Let $$\tau = (1,2,3)(4,5,6,7,8) \in A_{11},$$ where $A_{11}$ is the group of even permutations. Let $H$ be the centralizer of $\tau$. I can easily prove that $H$ is a subgroup of $A_{11}$. How do I ...
2
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1answer
46 views

For which $p$ is $G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$ a subgroup of $S_{12}$?

Let $p$ be a prime number. I've to figure out for which $p$ $$G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$$ is a subgroup of $S_{12}$. I realize that we have to ...
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1answer
12 views

Finding the smallest exponent $k$ for a non-cyclic permutation $\sigma$, so that $\sigma^k = id$.

What I am aware of (1) A cyclic permutation is a permutation that consists of a single nontrivial cycle (cycle of length $> 1$). Let $k$ be the length of the cyclic permutation $\tau$. Therefore ...
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3answers
78 views

Understanding the definition of the sign of a permutation , $\operatorname{sgn}(\pi) = (-1)^k$ .

I am trying to understand the definition of the sign of a permutation $\pi$. My textbook only mentions that $\operatorname{sgn}(\pi) = (-1)^k$ , where $k$ is the number of transpositions . But I ...
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1answer
23 views

Summation of series with binomial coefficients

The value of $$\sum {n\choose n-r} (n-r) \sin(r\cdot \pi/n)$$ where $r\in (0 ..,n)$ is equal to? I think the question can be solved by writing the series in reverse order but I am not able to solve ...
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1answer
42 views

Sets and probability

The number of total outcomes of an experiment are $25$. If $A$ and $B$ are two non-empty independent events of the experiment such that outcomes in favour of event $A$ are $15$, then the minimum ...
2
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1answer
57 views

Proper subgroup of $S_{15}$ that strictly contains $\sigma $

How does one prove that: There exists no cyclic proper subgroup of $S_{15}$ that strictly contains the following permutation (of order $10$)? $$\sigma = (1, 2,3, 4,5, 6, ...
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9 views

Why are two transitive actions of a group G equivalent if there exist an automorphism of G swapping two point stabilisers?

Let $G$ be a group acting transitively on two sets $\Omega_{1}$ and $\Omega_{2}$. Also let $w_{i}\in\Omega_{i}$ and suppose there exists $\alpha\in Aut(G)$ such that $\alpha(G_{w_{1}})=G_{w_{2}}$, ...
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2answers
29 views

Divisors of $75600$ of the type of $4n+2$

Find the total no. of divisors of $75600$ which of the type of $4n+2$ where $n\in \mathbb{N}$ and $75600=2^4 \cdot 3^3 \cdot 5^2 \cdot 7^1$ Now I think divisors of type $(4n+2)$ should be of ...
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1answer
30 views

Number of permutations with given cyclic structure

If $\sigma$ is a permutation made up by the disjoint cycles $\tau_1, \dots, \tau_r$ (including those of length $1$), we call structure of $\sigma$ $$(l_1, \dots, l_r),$$ where $l_1, \dots, l_r$ are ...
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1answer
47 views

Prove that the set $U = \{(123), (124), … , (12n)\}$ can be used to generate $A_n$.

A hint is provided with the proof prompt: $(abc) = (1ca)(1ab)$, $(1ab) = (1b2)(12a)(12b)$, and $(1b2) = (12b)^2$. My idea: $(1ab) = (12b)(12b)(12a)(12b)$. To solve for the other half of $(abc)$, I'm ...
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1answer
28 views

If $\sigma$ is a product of $k$ transpositions, prove that $k \equiv$ inv $(\sigma)$ (mod $2$).

An inversion in a permutation $\sigma = \sigma_1...\sigma_n$ is a pair of indices $i < j$ such that $\sigma_i > \sigma_j$. Let inv($\sigma$) be the number of inversions of $\sigma$. If ...
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2answers
12 views

Possible orderings when the items are not unique?

First of all, I'm sure this question has been answered somewhere on the web, but I am just starting probability and I don't have the vocabulary to know what to look for, which is why I am asking here. ...
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0answers
33 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
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4answers
37 views

Number of arrangements in which no two persons sit side by side

There are $10$ seats in the first row of the theater out of which $4$ are to be occupied. Find the number of ways of arranging $4$ people so that no two people sit side by side. Making different ...
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15 views

All equivalent moves on a rubik's cube

Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively. Let us say that I have a permutation of the stickers on the cube written as a word ...
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1answer
57 views

Subgroups of $G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}$ [closed]

Let $$G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}.$$ Is this a subgroup? By examining particular examples, one can see that it is not, since it is not closed under composition. However, ...
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29 views

Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group ...
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1answer
49 views

Order and generators of intersection of cyclic groups

Let $\sigma$, $\tau$ be two permutations of $S_n$. We know that $\langle \sigma \rangle \cap \langle \tau \rangle$ is a cyclic subgroup and (from Lagrange's Theorem) that its order divides ...
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31 views

What is the probaility that two random permutations have same order?

I am interested in the orders of random permutations. Since the law of the log of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), one ...
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1answer
38 views

The probability of being dealt at least 5 wanted cards

In a fictional deck of cards, there are 30 cards, 15 different ones (each card has an identical pair, so 15 pairs = 30 cards). I want to answer the question: I am dealt 10 cards. I wish to receive 5 ...
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1answer
34 views

Subgroups of $S_{10}$ that contain $\langle \sigma = (123)(456)\rangle$

In $S_{10}$ let us consider the permutation $\sigma = (123)(456)$ and the cyclic subgroup $ G = \langle \sigma \rangle$. I can fairly easily find that $$\langle \alpha = (1,4,2,5,3,6)(7,8) \rangle ...
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1answer
19 views

Multi-stage Probability

I think the easiest way to explain what I'm having trouble with is to give an example question: A monkey is given 12 blocks: 3 Squares, 3 Rectangles, 3 Triangles, 3 Circles. Calculate the probability ...
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1answer
15 views

find all combination without overlapping

(*Constraint) Kinds of number should be limited in 3 (ex. {1,1,1,1} o/ {1,2,3} o / {1,1,2,2,5,5,4} x) And I want to find series of Integer. For example, if n = 4 (n is length of numbers) {2, 7, 2, ...
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1answer
24 views

Probability of drawing $m$ of $A$ in $n$ cards given a deck of $d$ cards contain $a$ copies of $A$?

As in the title I'm trying to work out what the chances of drawing $m$ copies of a specific card in $n$ draws are given a deck size of $d$ containing $a$ copies of $A$. I've tried using permutations ...
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2answers
31 views

Find the number of seating arrangements at a round table of three single men, two single women, and two families

Three single men, two single women and two families take their places at a round table. Each of the two families consists of two parent and one child. Find the number of possible seating arrangements. ...
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1answer
19 views

Number of permutations of $S_n$ such that $\sigma^h(a) = \sigma^k(b)$

A basic result in combinatorics is: In $S_n$ there are $(n-d)(n-2)!$ permutations $\sigma$ such that $\sigma^k(a) = b$, if $a \neq b$; $d(n-1)!$ permutations $\sigma$ such that ...
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2answers
42 views

In how many ways can a committee of $6$ people be selected from $7$ men and $6$ women if it can contain at most one of persons A and B?

A committee of $6$ people will be formed with $7$ men and $6$ women. The oldest of the $7$ men is A and the oldest of the $6$ women is B. It is described that the committee can include at most one of ...
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34 views

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$ $H$ is the set of fixed points on $A$ $A$ : set, $H \le S_A$, $F(H) = \{ a \in A : \sigma (a) = a, ...
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1answer
21 views

Prove the set of permuations which permute only finitely many elements is a normal subgroup

Let $A$ be a non-empty set and let $X$ be a subset of $S_A$ Now let $F(X) = \{a \in A : \sigma(a) = a, \forall \sigma \in X\}$, $M(X) = A\backslash F(X)$, and $D = \{ \sigma \in S_A : \mid ...
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2answers
23 views

Probability; bridge hand question

$13$ cards are chosen at random with no replacement from a deck of $52$ cards. find the probability there are $5$ spades chosen, $4$ hearts, $3$ diamonds and $1$ club. I got ...
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1answer
34 views

Properties of lists with arbitrary lengths and alphabet size

I am having trouble understanding a problem of applying the concepts of permutations and combinations in an example that I found while reading my textbook. Basically, it wants the number of elements ...
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3answers
33 views

How many different varieties of pizza can be made if you have the following choices:

How many different varieties of pizza can be made if you have the following choice: small medium, large; thin, hand tossed, pan; and $12$ toppings (cheese is an automatic), from which you may select ...