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

learn more… | top users | synonyms (1)

1
vote
1answer
26 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 "mediterranean"...
0
votes
1answer
50 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 $f(...
0
votes
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
36 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
votes
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
votes
1answer
17 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
votes
2answers
52 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
votes
2answers
28 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
votes
1answer
24 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
votes
0answers
22 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
votes
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
vote
1answer
19 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
votes
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 ...
1
vote
2answers
38 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 ...
1
vote
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
43 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 a ...
2
votes
1answer
44 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
votes
2answers
36 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 $e$ of $...
1
vote
0answers
12 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
votes
0answers
24 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
vote
2answers
31 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
25 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 ...
6
votes
2answers
207 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
37 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
votes
2answers
36 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
votes
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 ...
3
votes
1answer
39 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
votes
0answers
36 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 ...
1
vote
2answers
119 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 $\lfloor{\...
1
vote
1answer
52 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 ...
1
vote
0answers
33 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
153 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
vote
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
28 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 $n$...
0
votes
0answers
15 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 ...
2
votes
1answer
57 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, \operatorname{sgn}...
1
vote
1answer
64 views

$10$ people are standing in a queue when three new checkouts open. In how many ways can the three new queues be formed?

Problem: $10$ people are standing in a queue when three new checkouts open. 8 people rush to the new checkouts and the new queues end up with at least two people in each. In how many ways can the ...
1
vote
3answers
39 views

Probability in Permutation [closed]

A permutation of $1,2,3,\ldots,n$ is chosen at random. Then the probability that the numbers $1$ and $2$ appear as neighbours equals ______? Options are A) $\dfrac 1 n$ B) $\dfrac 2 n$ C) $\dfrac{1}...
0
votes
1answer
43 views

Prove the pullback of the wedge product is the wedge product of the pullbacks.

Let $F:V \rightarrow W$ be a linear map. Show that $F^{\ast}(\omega \wedge \eta)=(F^{\ast}\omega) \wedge (F^{\ast}\eta)$ for all $\omega \in \Lambda^{p}(W) , \eta \in \Lambda^{q}(W)$. Where $F^{\ast}...
3
votes
2answers
66 views

math software - permutation group elements operation

I need a software allows to calculate operation elements of permutation group. For example the following elements operation yields identity permutation $$ (1234)(1423) = (1)$$ Sage seems to solve the ...
1
vote
0answers
29 views

Trouble understanding Sylow's Third Theorem

The statement of Sylow's third theorem in my text goes like this, Let p be a prime and let G be a group of order $p^km$, where $p$ does not divide $m$. Then the number $n$ of Sylow $p$-subgroups of $...
0
votes
0answers
30 views

Modifying permutation function for inputs with equivalent ratios

I have the following function: $$f(a_1,a_2,\ldots,a_n) = \frac{(a_1 + a_2 + \cdots +a_n)!}{(a_1! a_2! \cdots a_n!)}$$ where $a_i\ge 0$ I need to modify this function such that $f(a_1,a_2,...,a_n) = ...
0
votes
0answers
37 views

Different ways in which a micro-switch with eight switches can set

A computer interface for a Kawai digital studio piano has eight micro-switches that can be set in either the "on" or "off" position. These switches must be set properly for the interface to work. In ...
-1
votes
2answers
48 views

How to calculate the number of non decreasing functions between two finite sets? [duplicate]

I want to know how to calculate number of non decreasing functions from one set to another set. Let $A=\{1,2,3,\ldots,10\}$ and $B=\{1,2,3,\ldots,25\}$ Please tell me an easy method to calculate the ...
1
vote
2answers
54 views

Bernoulli Numbers and Tangent numbers.

Good evening. I am looking to see if there is a proof online to help guide me with the understanding that the Tangent Numbers, denoted $T_n$ and the Bernoulli numbers, denoted $B_n$ are related. It ...
1
vote
2answers
80 views

How many permutations can be formed from $2n$ distinguishable objects and $n$ indistinguishable objects?

How many permutations can be formed from $2n$ distinguishable objects and $n$ indistinguishable objects? Please tell me if I am on the right track to solving this question. Basic Formula: $C(n+r-1,...
-2
votes
1answer
25 views

Generate all Permutations of Four Events, Three Outcomes each

Hello I would like a list of all permutations for the following set up. I tried an online permutation generator, but I didn't quite get it working, so I'll try this forum, which has been great in the ...
0
votes
0answers
22 views

A probability word problem

There is an infinite rectangular window, with infinite vertical iron bars dividing it at distances of 30 centimetres. Now, a 15 cm long pen is randomly thrown at the window. Assuming that the pen can ...
1
vote
1answer
36 views

Is an orthogonal matrix necessarily a permutation matrix?

Is an orthogonal matrix necessarily a permutation matrix? I believe the answer is no as a permutation matrix is a special case of an orthogonal matrix, but I am having a trouble finding a ...