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

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4
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2answers
60 views

number of pairs formed from $2n$ people sitting in a circle

I am trying to understand the solution to the following problem: Suppose that $2n$ persons are sitting in a circle. In how many ways can they form $n$ pairs if no two adjacent persons can form a ...
0
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1answer
15 views

Probability : Container arrangement

There are a total of 15 containers out of which two containers have same color and the remaining are of different colors. The question is to find the probability that i) Two containers with same ...
1
vote
1answer
39 views

How should I continue my proof of this cycle property? (And did I make a mistake?)

I am trying to show: For a given single cycle, such as $(1, 4, 5, 7)$, the order of such a cycle is the length of the cycle. (i.e $(1, 4, 5, 7)^4 = ()$). I am trying to do this by induction. ...
1
vote
3answers
68 views

How to count permutations with cycles of length at least 51 in $S_{100}$?

Let consider permutation $ \in S_{100} $ How to count the number of permutations of those which contains a cycle of length 51 at least. ( so I would like a cycle of length 52,53,54,....,100)
0
votes
1answer
46 views

A simple question about permutations [closed]

So I could not find an answer anywhere, so here it is: If a string could be consisted of x y x y x y x y and x could only be used once, while y could be repeated, would it be correct to say that ...
0
votes
2answers
23 views

Combination and Permutation Math Problem

I am having some difficulty dissecting this problem and solving it: The track team has 7 girls and 6 boys. For the meet next week, they must choose a runner, a pole-vaulter, a captain and a ...
0
votes
1answer
19 views

Square labeled with same number.

Recently I met this combinatorics problem: "Let all points with integer coordinates in a plane be labeled with one of the numbers $1,2,3,...,n$. Prove that there is a rectangle whose vertices are ...
0
votes
2answers
14 views

How to maximize sum of pairwise multiplication of array elements taken one from each array?

Suppose you are given two arrays: $$a = [a_1,a_2,a_3,\dots,a_n],\hspace{5mm} b = [b_1,b_2,b_3,\dots,b_n]$$ Now you need to take one element from $a$ and one from $b$ multiply it, and add to sum and ...
0
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0answers
23 views

Form of an element of a normal subgroup of $A_n$

I want to show that $A_n$ is simple for $n\geq 5$. For $n=5$ I have used the following criterion Let $H$ be a normal subgroup of $A_5$ then $H$ can contain any one of the following $a.$ a $5$ ...
0
votes
0answers
24 views

Possible combinations of N different balls in M identical bins with the same capacity L

For a distribution center I am interested in the number of possible combinations to put N different boxes (all the same size but different content) into M equal bins (containers) with capacity L, with ...
0
votes
1answer
54 views

How many different permutations?

Suppose I've n boxes and m different colored balls of different quantities.How many unique permutations can be obtained ? Example : n=2,m=2, with quantities ( A - 1 ball, B - 2 balls) Thus the ...
0
votes
0answers
28 views

What is sgn(321)?

I've tried to compute the length of (321) and I got 2. Then the sgn should be (-1)^2=1. But I suppose sgn(321)=-1 by the definition in the graph?
2
votes
0answers
30 views

Please check my solution of a problem in combinatorics regarding partitions

A lift automatically operated has a further computer facility of recording how many people leave the lift at each floor. It starts at floor $1$ and goes up to floor $6$. If $8$ people consisting of ...
4
votes
1answer
33 views

Permutation question of arranging people in a row

There are $6$ boys and $4$ girls in a class. How many ways are there to arrange them in a row if no girl stands next to each other? I would know how to solve this if there are only $2$ girls. But ...
0
votes
1answer
20 views

Probability distribution of number of ordered items in a permutation

I have a simple algorithm to check if a series of numbers is sorted: if the first two numbers are sorted, move to the next two. Else, stop and return false. I want to figure out the average case ...
3
votes
1answer
66 views

To determine number of arrangements of 4 letters in a word so that the transitions remains conserved

A 10 letter word is composed of $A,\ B,\ C,\ D$. The problem is to find the number of arrangements of these alphabets which could lead to fixed number of transitions between each pair of alphabets. ...
0
votes
0answers
46 views

Prove whether or not a group is cyclic [duplicate]

I need to decide whether or not the group $\langle(12)(34)(56), (145)(236)\rangle \leq S_6$ is cyclic; if it is, I must find its generators, and if it isn't, I must prove that no generators exist. I ...
-2
votes
1answer
25 views

How do you do this permutation? [closed]

How would you do this math problem? and how do you know to use the equation you use? Mendy’s offers three types of bread: White, Whole-Wheat and Rye. The choices of meat are corned beef, pastrami, ...
0
votes
3answers
42 views

Subgroups of a permutation group

The permutation group $S_{4}$ is defined as the group of all possible permutations of [1234]. i) Find the number of subgroups of $S_{4}$ that have order 2. ii) A: { [1234], [2143], [3412], [4321] } ...
2
votes
3answers
108 views

prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$ [duplicate]

How can i prove that : $$\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$$ i tried to prove it : by $$\binom{2n}{n} = \frac{(2n)!}{n!n!} = \frac{2^n ...
3
votes
2answers
50 views

How to prove this equality for Stirling numbers?

How can I prove that the following formula is true for Stirling numbers of first kind. $$\sum_{k=1}^n(-1)^k\left[\begin{matrix} n\\k\end{matrix}\right] =0$$ Actually I want to prove that number of ...
0
votes
0answers
16 views

Counting with permutations and counting ignoring permutations

I am given this problem: This problem was given to me in my computer science class but it has to do with permutation and I want to understand it mathematically first. let $c(n)$ be the number of ...
0
votes
1answer
63 views

Four Letter-envelop problem

A secretary writes four letters and the corresponding addresses on envelopes. If he inserts the letters in the envelopes at random irrespective of the addresses, (i) find the probability that only one ...
6
votes
2answers
33 views

Number of visible elements in a permutation

The following problem occurred to me the other day, and I've played around with a bit but can't seem to find a good solution: Consider a permutation $\pi$ of $\{1, 2,\ldots ,n\}$. For every positive ...
0
votes
0answers
17 views

probability of vector in column span

Consider we have a fixed matrix M of size a$\times$ 2b (Let us look at M=[$M_1$ $M_2$] where matrices $M_1$,$M_2$ are of size a$\times $b) and a vector $v$ of dimension a. Is there any way that I can ...
7
votes
1answer
99 views

Outer automorphisms of the infinite symmetric group

Denote by S$_\infty$ the group of permutations of $\mathbb N$. Question: Does there exist an outer automorphism of S$_\infty$, and if so, can one be exhibited? Does this depend on the continuum ...
0
votes
1answer
47 views

Generators of symmetric group $S_n$ [duplicate]

How can you prove that $S_n$ is generated by $(1\space 2)$ and $(1\space 2\space 3\space ... \space n))$ for $n\geq 2$?
0
votes
1answer
14 views

Finding cycles with set of permutations

Let $\alpha = (\alpha_1 \, \alpha_2 \, \ldots \, \alpha_s)$ be a cycle, for positive integers $\alpha_1 , \alpha_2 , \ldots , \alpha_s$. Let $\pi$ be any permutation. Show that $\pi \alpha \pi^{-1}$ ...
0
votes
0answers
17 views

the number of permutation $p$ of $A$ such that $\forall i\forall x\in A_i \,\,\, p(x)\notin A_i $

Given a collection of finite sets $(A_i)_{i=1,...,n}$, Let $A=A_1\cup\cdots \cup A_n$, my question is : What is the number of permutation $p$ of $A$ such that $\forall i=1,\cdots,n$ we have ...
3
votes
1answer
45 views

Does the product of all conjugates of some subgroup is independent of the order?

Let $G$ be a finite group and $A \le G$. Let $A^G = \{ A_1, A_2, \ldots, A_n \}$ be all the conjugates of $A$, i.e. each $A_i$ equals $A^g$ for some $g \in G$. Then I want to show that $$ A_1 A_2 ...
0
votes
1answer
22 views

permutations of 10 objects in a subset contains similar elements

A board that is divided into 15 different places, and we want to place 10 components on this board such that each component is placed in a section; knowing that those components are divided into 4 ...
0
votes
3answers
62 views

Simple Probability Question about Combinations

If someone could please point me in the right direction on these. I get lost on how to think about them. In a game there are four holes with values 0, 1, 2, and 4. You are given 6 balls to shoot into ...
0
votes
1answer
20 views

Double Check Probability for Permutation

I have to find the sample space and a few probabilities here and I am wondering about if I am going down the right track for these. If I am incorrect, then please point me in the right direction, but ...
1
vote
2answers
27 views

Question about permutations: How to show $\sigma(P)=(-1)^{\imath(\sigma)}P$?

A permutation of a finite set $X$ is any bijection from $X$ to $X$. We denote by $S(X)$ the set of all permutations of $X$. If $I_n:=\{1, \ldots, n\}$ we write $S_n$ instead of $S(I_n)$. Define ...
1
vote
2answers
26 views

Parity of permutation example

I know the definition of parity of permutation. But what does that look like in examples? For example, if the number of permutations is odd, then the sign of permutation in $-1$. What does this mean? ...
1
vote
2answers
35 views

24 possible combinations?

I'm terrible at math. My goal was to try and create $24$ total combinations using a horizontal bar, and a horizontal bar with one break. So that's two bar types. How can I write out the math that ...
1
vote
1answer
17 views

How many different arrangements of these 7 dogs are there if no poodle is next to another poodle?

Question: 2 Spaniels, 2 retrievers and 3 poodles go through to the final. They are placed in a line. How many different arrangements of these 7 dogs are there if no poodle is next to another poodle?
0
votes
1answer
46 views

Number of possible passwords - Google APAC Test

I am trying to solve this problem - Google apac test - Password Attacker . Problem summary: Using $M$ distinct characters, what is the number of ways of making a password of length $N$, such that ...
0
votes
1answer
23 views

Example of an elementary permutation

Can someone please give an example of an elementary permutation? The book says that every permutation can be written as a composite of elementary permutations. Can someone please give an example? ...
0
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0answers
26 views

Count number of trees

Given an array with n elements which is the pre-order traversal of a tree. How many max-heap will have the same pre-order traversal?
3
votes
1answer
81 views

Arrangement of integers in a row such that the sum of every two adjacent numbers is a perfect square.

Inspired by this interesting question and in order to solve an old problem, I have the following question: Can we construct a strictly increasing sequence $(N_i)_{i\in \mathbb{N}}$, such that for ...
0
votes
1answer
41 views

Murder mystery permutation problem

In exploring a hypothetical situation, I ran across this problem and I'm curious to know the answer, but math's not really my forte. You have a pool of 15 people. Between these 15 people there will ...
1
vote
2answers
53 views

How to find the number of strings of length N that can be formed by using the characters A, B, and C only that do not have “ABC” as a substring?

A, B, and C can be used any number of times in the string. This problem appeared in a programming contest which is already over. http://www.codechef.com/problems/CDSW152
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1answer
28 views

What kind of mathematical approach can you use to find all non-repeated combinations?

At first glance I thought this was a non-repeated combination or permutation, but those use a set length. So, I guessed this might be a partition of a positive integer, but it's not looking like that. ...
4
votes
1answer
42 views

An Example for a Graph with the Quaternion Group as Automorphism Group

I am reading "Graphs of Degree Three with given Abstract Group" (by Robert Frucht) where the author describes (somewhat tedious) algorithms to construct suitable graphs starting from a given group. I ...
2
votes
0answers
19 views

How to calculate total results of combinations of letters

I am programmer and have developed an algorithm to run a processor intensive function on all the permutations of 2 letters (X and O) when we define how many X's and O's there will be. For example, I ...
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0answers
30 views

Structure of the semidirect product decomposition

I'm looking at a complicated group that involves many semidirect products, and I realized that I have a fundamental confusion about how to use the structure of a semidirect product decomposition of a ...
1
vote
1answer
39 views

Possible permutations of a grid

I hope this is the correct place to post this, as I don’t study maths. But I do need help calculating the possible permutations of a grid based game I’m currently programming. This isn’t to help out ...
0
votes
1answer
38 views

Find Number of combinations possible

There are two letters "X" and "Y".A String of length N needs to be formed using those two letters How many number of combinations that can be possible where N should start with "Y" and no two or more ...
0
votes
0answers
29 views

Number of ways to color N objects in X colors where there is at least one object of each color.

What is the number of way to color $N$ objects in $X$ colors, where there is at least 1 object of each color?