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

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2
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1answer
13 views

Permutations starting with a specific letter

Ok, this is a homework question and I think I've resolved it but I want to bounce it off you guys. I have a $6$ letter word with no repeated letters. I need to calculate how many $3$ letter words can ...
2
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0answers
38 views

Positive integers $<100000$, how many contain exactly one $3$, one $4$ and one $5$

So I use $5$ positions for range $00000$ to $99999$ Choose $3$, choose $4$ and choose $5$ as follows: $5C1 \cdot 4C1 \cdot 3C1$ Remaining $2$ digits have $7$ possible digits as input Ans: $5C1 ...
3
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1answer
43 views

Painting a 2x2 Grid

We have a 2x2 grid and 10 different colours. I want to paint such that adjacent grids are painted with different colors. How many ways can i do this? ...
2
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2answers
30 views

$5$ chem students, $6$ maths students and $7$ physics students permutation

$5$ chem students, $6$ maths students and $7$ physics students. Find the number of arrangements if a)Chem majors are to occupy the first 5 positions b)Chem majors cannot occupy the first 5 positions ...
1
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0answers
11 views

Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph

I am trying to understand Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph. I quote it from Cycles in graphs and groups by Kantor. Theorem $1$ ...
8
votes
1answer
63 views

Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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2answers
35 views

Finding the parity of a permutation “exclusively”?

I'm trying to find the parity of permutations such as $(2468)$. What makes it possible to find the "exclusive" parity of such permutation? I.e. that if one tries to express $(2468)$ as a product of ...
0
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2answers
28 views

If $\sigma=(a_1 a_2 … a_n)$ and $|\sigma|$ is odd, then what is $\sigma^2$?

I'm trying to understand the way to infer the power of a permutation. If $\sigma=(a_1 a_2 ... a_n)$ is a $k$-cycle and $k=|\sigma|$ is odd, then how can I infer what $\sigma^2$ is?
0
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1answer
69 views

Prove $sgn(π) = sgn(π^{-1})$?

I'm pretty sure the inversion count of $π$ should be the opposite of the inversion count of $π^{-1}$. By this I mean if $π$ looks like this: $1 \to 1$, $2\to 2, \ldots, 10 \to 10$ and therefore the ...
0
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1answer
27 views

Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
1
vote
1answer
32 views

Write $π = (3, 2, 5)(2, 5, 4)$ in “table” notation?

Isn't this impossible...? Because this permutation goes from 3 --> 2 ---> 5 ---> 3 according to the first cycle, but goes from 2 --> 5 ---> 4 ---> 2 according to the second cycle. So 5 can't go to 3 ...
1
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1answer
18 views

Permutation of the alphabets of the word “mediterranean” such that first and fourth letter are “r” and “e” respectively.

Above is the original question. The correct answer is in green that is 59. I have chosen option 3 that is $\frac{11!}{(2!)^3}$ because I thought that there are 13 alphabets in the word ...
0
votes
1answer
48 views

Prove that the funtion f: $G\rightarrow G$, defined by $f(x)=x^k$, $x \in G$ is a permutation of $G$

Help me with this exercise, I could not do it :( Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove that the function f: $G \rightarrow G$, defined by ...
0
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0answers
17 views

Show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$

Here's the full question: If $σ ∈ S_n$ is any permutation and $i_1, . . . , i_k $ are $k$ distinct elements of $\{1, . . . , n\}$, show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , ...
0
votes
1answer
32 views

How can we compute restrictions from a character table?

I would like to how to, when given a character table, calculate the restriction. $Res_H^G : Rep(G) \rightarrow Rep(H)$. For example: Let $G=S_4$ whose character table is given below (see ...
0
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0answers
22 views

Need help calculating number of possible passwords with given criteria

I need help calculating the number of possible passwords with a given set of criteria. Here is the set of criteria: Passwords are case insensitive. Must be 6-14 characters. Must contain at least 1 ...
0
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1answer
14 views

Counting monomials with $k$ variables

Say we expand $\left(\sum_{i=1}^n x_i\right)^k$ into monomials. If $k=3$ there are $3n(n-1)$ monomials with two variables: $3x_1x_2^2 + 3x_1x_3^2 +\dots + 3x_1^2x_2 + \dots$. Is there a closed form ...
0
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2answers
44 views

Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$.

$S_n$ is the set of all permutations. Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$ I can somewhat see why this statement would be true, seeing as permutations are read from right to ...
2
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2answers
27 views

Suppose $π ∈ S_n$, and for this $π$ define $C_π : S_n → S_n$ be defined by $C_π(σ) = πσ$. Why is $C_π$ a bijection?

$S_n$ is the set of all permutations. I'm just starting on this material, so I'm confused on how to read this problem. Does the function consist of multiple permutations (i.e. the permutation of a ...
-2
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0answers
19 views

Prove that pn = 1 n × pn−2 + n − 1 n × pn−1

Prove that p_(n) =(1/n)× p_(n−2) + ((n − 1)/n) × p_(n−1). if the first element in this permutation is j != 1 and we do 1) jth entry is 1 2) jth entry is not 1.
2
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1answer
20 views

Find The Number Of Outcomes

I understand how to find the number of outcomes using permutations and combinations, but then I thought to myself what happens when it involves both? I will make a mock scenario to explain what I am ...
0
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0answers
20 views

the length of the conjugate class containing $\alpha$ in $S_n$ [duplicate]

Suppose $\alpha$ $\in$ $S_n$ and there are exactly $n_i$ $l_i$-cycles ($i=1,2, ... ,k)$ (containing $1$-cycles) in the cycle decompostion of $\alpha$ ., then the length of the conjugate class ...
2
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1answer
21 views

Converting Permutations to Combinations: Simple Stats in Practise

In a popular text book there is a question that has bothered me that I am sure is very simple for others and I'm just missing something..... So image $100$ songs and we have $10$ as Beatles songs. We ...
1
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1answer
11 views

Generate a unique combination from an index within the number of combinations

I'm writing a program which will use a genetic algorithm optimize neural networks to play tic-tac-toe (That's not related), and I've come across the following problem: I'm looping through every ...
0
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1answer
20 views

Permutation : Is there any formula to solve this?

Given, 14 objects of type A 8 objects of type B 3 objects of type C 2 objects of type D Find the permutation of 10 objects? Is there any general formula in permutation to solve a problem ...
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0answers
12 views

Combinations OR Decision tree ? Six Spices - Total flavors

this is a simple question for which I'm trying to reason. Suppose you have 6 spices, what is the possible number of flavors you can make ? You may assume that you can only combine one spice once to ...
1
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2answers
32 views

What is an intuitive explanation of the combinations formula?

I perfectly understand the permutations formula i.e. if you have $n$ things how many ways can you rearrange it if taken $k$ at a time (or if you have $k$ slots)? So you draw the following tree. And ...
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0answers
27 views

Is $N_{A_7}(H) = H$, with the following $H$?

I am following a proof in which I have a subgroup of $S_7$ defined by $H := \langle (2, 3, 4)(5, 6, 7) , (2, 7, 6, 3)(4, 5) \rangle$ The book implicitly uses that $N_{A_7}(H) = H$ (the normalizer ...
2
votes
1answer
50 views

About conjugating a $7$-cycle in a subgroup of $S_7$

Following a proof in which I have a transtive group $G$ of order $168$ , which is a subgroup of $S_7$ (I am trying to characterize it, I cannot use well know facts such as it is always isomorphic to ...
1
vote
1answer
42 views

Proof to an observation of stabilisers and orbits

Observation: If $\alpha^{g}=\beta$ then $G_{\beta}=g^{-1}G_{\alpha}g$ Just to get the notation out of the way: $G_{\beta}= g^{-1}G_{\alpha}g$ is the stabiliser of a point element $\beta$ in ...
2
votes
1answer
37 views

Number of Elements in a Conjugacy Class of $S_N$ (Derivation)

Consider the conjugacy classes of the symmetric group $S_N$. Each conjugacy class consists of permutations that have the same cycle structure. We see that the number of possible cycle structures is ...
3
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2answers
31 views

An automorphism that has no fixed points except for the identity and is its own inverse implies commutativity

Let $G$ be a finite group and suppose there exists $f\in\text{Aut}(G)$ such that $f^2=\text{id}_G$, i.e., $f$ is its own inverse, and such that $f$ has no fixed points other than the identity $e$ ...
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0answers
9 views

Removing the dimension factor in Fannes inequality

Given two distributions $x=(x_1,\ldots, x_n),y=(y_1,\ldots y_n)$ on $[n]$, it is known by Fannes inequality that $H(x)-H(y)\leq O(\|x-y\|_1\log n)$, where $H(\cdot)$ and $\|\cdot\|_1$ represent ...
0
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0answers
23 views

Derangements One to One Functions

What is the number of one to one functions from $\{1,2,3,\cdots,n\}$ to $\{1,2,3,\cdots,n\}$ so that $f(x)\neq x$ for all $x$. I understand that $A_1 \cup A_2 \cup A_3 \cup\cdots\cup A_N$ is the set ...
1
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2answers
28 views

Decomposition of permutation

I was asked to decompose the permutation $$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1 \\ \end{pmatrix} = (12345) \in S_5$$ into a product of two ...
0
votes
1answer
21 views

Polynomial roots conditions vary with coefficients

Polynomial equation $\sum_{i=0}^4 p_i x^i=0$ have the following root conditions: 1) $a_0 \pm b_0 i, a_1 \pm b_1i$ 2) $a_0 \pm b_0 i, a_1, a_2$ 3) $a_0, a_1, a_2 \pm b_2i$ 4) $a_0, a_1, a_2, a_3$ I'm ...
5
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2answers
193 views

What is an intuition behind conjugate permutations?

I know the definition of conjugate permutations. $$\exists p \quad p^{-1} \alpha p=\beta$$ So the $\alpha$ and $\beta$ is a pair of conjugate permutations. But can anybody can give some concise, ...
0
votes
2answers
35 views

Calculate the centre $Z(G)=\{g\in G| gx=xg \forall x\in G\}$.

Let $G$ be the symmetry group of a square. The first exercise is that I describe $G$ as a permutation group. I've found the following permutations: $ \begin{align*} \tau_1 &= \begin{pmatrix} 1 ...
0
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2answers
33 views

Generalized prisoners' problem

I am trying to generalize the prisoner's problem. The problem reduces to this: find the probability that a random permutation of $1,...,n$ has no cycle of length $>L$. If the number of ...
0
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2answers
61 views

How is $\frac{n!}{(n-r)!}$ the same as $n(n-1)(n-2)…(n-r+1)$

I'm a bit confused about permutations in my stats course: How is $\frac{n!}{(n-r)!}$ the same as $n(n-1)(n-2)...(n-r+1)$ I'm use to learning it permutations being $\frac{n!}{(n-r)!}$ but not the ...
2
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0answers
30 views

Permutations of $S_7$

Find all permutations $\alpha \in S_7$ such that $\alpha^3 = (1 2 3 4)$. My attempt: We know that such an $\alpha$ must "look like" $(1432)$, since $(1432)^3=(1234)$. I think I need to find the ...
0
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0answers
34 views

Finding a probability, related to sets of permutations

Let $\Omega$ be the collection of permutations of the set $\{1,2,...,n\}$ with the normalised counting measure: $$P(A) = \dfrac{\text{ number of permutations belonging to A }}{n!}$$ For each $i$, let ...
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2answers
106 views

What is probability that at least $2$ people have same birthday from group of $N$ people?

Question is not that simple. There are also leap year included.Leap year will be $366$ days and normal year will be $365$ days. There is a statement in question that : there are exactly ...
1
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1answer
51 views

how to find digits except trailing zeroes

I have came across many questions of permutations and combination but I am confused in these types of questions “how to find last two digits" except trailing zeros in $1000!$ where $!$ represents ...
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0answers
26 views

Solvability if two pieces of the fifteen puzzle are identical?

It's known that only half of all the permutations in the fifteen puzzle can be solved (in the sense of recovering the sequential order of numbers, with the empty slot in the lower right corner), for ...
3
votes
2answers
120 views

Permutations of numbers $1, 2, 3,\dots,n$

How many permutations do the numbers $1, 2, 3,\dots,n$ have, a) in which there is exactly one occurrence of a number being greater than the adjacent number on the right of it? b) in which there are ...
1
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0answers
12 views

What's uniform block signed permutations?

Let $[n]=\{1,2,\ldots,n\}$ and $P(n)$ the set of all partitions of [n]. A partition of $[n]$ is non-empty disjoint subsets of [n], called blocks, whose union is $[n]$. A block permutation of [n] is ...
0
votes
1answer
27 views

Filling k positions with objects from $n$ different types

There are $n$ different types of objects and $k$ positions where an object can be placed. How can I determine the number of ways in which these $k$ positions can be filled by using objects of these ...
0
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0answers
14 views

Matlab: How to find a permutation matrices

I'm trying to figure out a way to compute the permutation matrices R and L given two matrices A and B. I would like to get L and R given that I know A and B. B=L* A* R. I wrote the code below for ...
0
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0answers
15 views

Decompose induced representation of $S_2$ and $S_3$

Let $ H=S_2 \subset G=S_3 $. Then use Frobenius reciprocity to decompose $ \operatorname{Ind}_H^G(\operatorname{sgn}_H) $ into irreducibles. $ G=S_3 $ has $ 3 $ irreps $ 1_G, ...