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

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5
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243 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
46 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
20 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
56 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
32 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
37 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
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1answer
45 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
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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
15 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
votes
4answers
47 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
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2answers
31 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
votes
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 ...
2
votes
2answers
43 views

Writing $P_n=\sum_{\sigma \in \mathfrak{S}_n} X^{c_n(\sigma)}$ as irreducible factors in $\mathbb{Q}[X]$.

Let $\sigma \in \mathfrak{S}_n$, denote $\alpha_n(\sigma)$ the number of cycles in the decomposition of product of disjoint cycles. Let $$P_n=\sum_{\sigma \in \mathfrak{S}_n} ...
1
vote
1answer
43 views

Product of disjoint cycle example

$(123)(45)(15)(24) = (14)(235)$, according to my lecture notes, yet I keep getting $(143)(25)$. By doing $$1 \to 4\\ 4 \to 3\\ 3 \to 1\\ 2 \to 5\\ 5 \to 2$$ Where am I going wrong?
0
votes
2answers
27 views

Finding expectation and variance of a selection of three balls out of six?

I just asked this question, but worded it wrong so while the given answers are useful, they still leave me confused for where I am in the progression through my stats book. My problem is I've got 3 ...
0
votes
1answer
16 views

permutations with specified repetition counts [duplicate]

Problem: Determine the number of permutations of the characters for: AABBBC How can I calculate a problem like this generally, given a set of characters and a number of times each has to appear?
1
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1answer
21 views

Repeated Permutations

In school we have been studying combinations and permutations, and in a programming assignment I was testing points on a coordinate plane. Testing all integer points surrounding the point (0,0) you ...
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2answers
24 views

Permutation Problem need help [closed]

So there is 7 people seated at a circular table. Person A cant move. How many ways can they be seated If person A stays in their seat?
0
votes
0answers
60 views

Is this an action of $S_{n}$ on $\mathbb{R}_{n}$?

I am trying to prove that $S_{n}$ acts on $\mathbb{R}_{n}$ with the map $$* : S_{n} \rightarrow \mathbb{R}_{n}, \quad * \left( \sigma, \left( r_{1}, r_{2}, \dots, r_{n} \right) \right) = \left( ...
1
vote
2answers
43 views

Finding The Number Of Inversions In A Permutation

Let the be the following permutation: $(1 5 4)(3 6)\in S_6$ How do I count the number of inversions to calculate the sign of the permutation? $(1 5 4)(3 6)=(1 5)(1 4),(3 6)=3$ so it has an ...
2
votes
2answers
53 views

Number of strings [closed]

There are $2^{10} =1024$ possible $10$ -letters strings in which each letter is either an $A$ or a $B$. Find the number of such strings that do not have more than $3$ adjacent letters that are ...
4
votes
2answers
58 views

Number of ways in which a batsman can score 14 runs in 6 balls not scoring more than 4 runs in any ball.

Hello everybody my query is regarding the number of positive integral solution. In the sport of cricket, find the number of ways in which a batsman can score $14$ runs in $6$ balls not scoring ...
0
votes
0answers
52 views

How to find the no. Of non negative integral solutions of a equation

I want to find the no. Of non negative solutions of $X+2y+3z=n$ I know how to find the non negative integral solutions of the equations of type $X+y+z=n$ using dividers method that is assume that ...
-1
votes
2answers
105 views

Equivalent permutation representations.

The definition of Equivalent Permutation Representations that is defined in "A course in Theory of Groups" by Derek Robinson Suppose we have group $G$ has permutation representation on set $X$ and ...
1
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1answer
13 views

Show that each conjugacy class has a particular value for probability after k steps

I have a permutation group $S_n$ and am performing random transpositions on them. Now there will be a bunch of conjugacy classes as a result of that. P_k_s is the probability that after k ...
0
votes
0answers
17 views

number of permutations that have $i<j$ against the ones with $j>i$

Consider a set $A=[a_1,a_2,\dots,a_n]$ and its all possible permutations $P$. Select one permutation $\sigma=(\sigma(a_1),\sigma(a_2),\dots,\sigma(a_n)\ )\in P$ and consider a set of distinct pairs ...
1
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2answers
48 views

Sign Of Permutation That Is Written As C Different Cycles

Prove: if $\sigma\in S_n$ is a factorization of $c$ disjoint cycles then $\text{sgn} (\sigma)=(-1)^{n-c}$. We know the one cycle sign is $(-1)^{l-1}$ so $c$ of them is: ...
1
vote
1answer
29 views

How many $(r+1)$- subsets of $[n+1]$ have $(k+1)$ as their largest element?

Let $[n+1]$ be the set defined by $[n+1]=\{1,2,\ldots,n+1\}$. Call a subset of $[n+1]$ with $r+1$ distinct elements an $(r+1)$-subset. How many $(r+1)$-subsets of $[n+1]$ have $(k+1)$ as their ...
3
votes
1answer
85 views

A fair die is rolled nine times. What is the probability that 1 appears three times, 2 and 3 each appear twice, 4 and 5 once and 6 not at all?

A fair die is rolled nine times. What is the probability that 1 appears three times, 2 and 3 each appear twice, 4 and 5 once and 6 not at all? My approach is fairly simple. The dice is fair, so we ...
0
votes
1answer
24 views

For a given set of pairings in the 8-team basketball tournament,in how many ways can the top 3 positions in the final standings be filled?

The top 2 teams must be from different brackets. I couldn't understand the question.In the initial competition,8 teams are separated into 4 groups(with 2 teams each) to compete.And it will give 4 ...
0
votes
1answer
20 views

Why $c(a_1 a_2 … a_k)c^{-1}$ is the k-cycle $(c(a_1) c(a_2)… c(a_3))$?

If $a,b,c \in S_n$, why $c(a_1 a_2 ... a_k)c^{-1}$ is the k-cycle $(c(a_1) c(a_2)... c(a_3))$? (I need this to prove that two permutations are conjugate iff they have the same cyclic structure.)
0
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2answers
81 views

probability of picking up two m&ms of same color randomly

There are 3 red m&ms, 5 green m&ms, and 8 blue m&ms. If I pick two m&ms out randomly, what is the probability of me picking two m&ms of the same color? I'm not sure if this is ...
1
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1answer
149 views

Pet Store Problem?

Hi I answered the problems just wanted to verify if my approach was correct. Any suggestions appreciated. Question: A pet store has 6 puppies, 9 kittens, 4 lizards, and 5 snakes. a. If you select a ...
2
votes
1answer
286 views

Baseball Combinations Problem

Two part question (My work below). For both questions will use the orioles current roster: -Current orioles roster: 12 pitchers, 2 catchers, 5 in-fielders, and 6 out-fielders: Similar to the list ...
0
votes
2answers
201 views

Combinations and Probability Problem

So far I got up to part C and I think I have to maybe divide my answer from part B by some number but am totally confused on how to approach this question. There are 15 dogs in an obedience class. ...
3
votes
1answer
17 views

No normal subgroup of a subgroup of $S_n$ imply the subgroup is the one of even permutations or consists of two elements

The following is an old exam question from a n introduction to group theory course: Let $G$ be a proper subgroup of $S_{n}$, $n\geq3$. Prove that if $G$ does not have any non-trivial normal ...
0
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1answer
21 views

Prove that $L_7$ is a subgroup of $S_7$

Let $\sigma(v)$ denote the signature of the permutation $v$. Is the subset $L_7 = \{v\in S_7 : \sigma(v)=-1\}$ a subgroup of $S_7$? I am not sure I am proving it the right way. To prove that ...
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1answer
30 views

more how many different possibilities stuff in a row

There's 4 green m&m, 5 red m&m, 8 blue m&m, 10 yellow m&m. In how many ways can you line them all up in a row. I believe the answer is 25840847132100 from an online total combinations ...
0
votes
1answer
26 views

Show that a cycle of length $p$ and a cycle of length $q$ in $S_n$ are conjugate if and only if $p = q$.

Show that a cycle of length $p$ and a cycle of length $q$ in $S_n$ are conjugate if and only if $p=q$. First of all, I'm a bit confused about the meaning of '... are conjugate'. Does this mean that ...
1
vote
2answers
53 views

Shortest possible way to go from one corner of the city to opposite corner if a city has $n,m$ parallel roads from east - west & north -south?

Let us suppose there is one city which has $n$ parallel roads running East - West and $m$ parallel roads running North - South. Now let us take that the distance between every consecutive pair of ...
2
votes
0answers
68 views

Permutation problem with ordering persons in a line

We have the following problem: There are $p$ persons from each city. Consider $p \cdot n$ persons from $n$ different cities. The $p \cdot n$ persons stand in a line such that every person stands next ...
0
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2answers
54 views

How many parallelograms can be formed when a parallelogram is cut by $2$ sets of $n$ parallel lines?

A parallelogram is cut by two sets of n parallel lines parallel to the sides of the parallelogram. The number of parallelogram thus formed is..?? I think we can do it by combinatorics.. But I'm not ...
3
votes
2answers
58 views

14 pencils handed out to 6 people. Each person has at least 1 pencil. Person 6 no more than 3 pencils.

We have 14 indistinguishable pencils and we want to hand out all of the pencils to 6 people and we want everyone to get at least one pencil. However, we do not want person 6 to get more than 3 ...
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2answers
48 views

How many different ways can 14 pencils be passed out to 6 different people? Some people are allowed no pencils.

There are 2 questions that are very similar and I have the same answer to both but I don't think that's correct. Can you help me see the difference between the 2 questions. We have 14 ...
7
votes
5answers
437 views

What's wrong with my permutation logic?

The given question: In how many ways the letters of the word RAINBOW be arranged, such that A is always before I and I is always before O. I gave it a try and thought below: Letters A, I and ...
0
votes
1answer
25 views

Find the population size that maximizes the probability that two random samples of size $20$ will have exactly $2$ members in common

Ten fish are caught in a lake, marked, and then returned to the lake. Two days later 20 fish are again caught, 2 of which have been marked. (a) Find the probability of 2 of the 20 fish being marked ...
2
votes
2answers
47 views

What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?

It seems that there is a action by which $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$ permute the $p^2$ ordered tuples in $\mathbb{F}_p^2$. What is the map from the $2 \times 2$ matrices over ...
1
vote
0answers
41 views

Permutation as a product of generators of the permutation group

Let $G$ be a permutation group, generated by $g_1,\ldots,g_n$. And let $h$ be in $G$. Example: $G=\langle (12)(34),(123)\rangle$ and $h=(12)(34)(123)=(243)$ (reading the cycles from right to left, ...