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
25 views

Permutations, compositions and associativity properties

Let n be a postive integer, and let σ : {1, . . . , n} → {1, . . . , n} be a one-to-one and onto map. Then σ is called a permutation on n elements. The set of all permutations on n elements is denoted ...
1
vote
3answers
43 views

Is there any permutation $\tau\in S_7$ so that: $\tau^{4}=\sigma$?

Let $\tau$ be a permutation in $S_7$: $\ \sigma= \left( \begin{matrix} 1 & 2 & 3 & 4&5&6&7\\ 3 & 4 & 5 &6&1&7&2 \ \end{matrix} ...
3
votes
1answer
41 views

How many ways 5 different books be distributed among 5 students

I've seen this question in a book and can't figure it out correctly. Let 5 different books be distributed among 5 students. Suppose the books are returned and distributed to the students again ...
0
votes
0answers
24 views

Find a permutation $f\in G$ such that $h=fgf^{-1}$

I've been given two conjugate permutations $h,g \in G=S_{11}$ and have to find another permutation in $G$ such that $h=fgf^{-1}$. This seems similar to a change of basis for a matrix which I can do ...
5
votes
4answers
220 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
vote
2answers
27 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
17 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
48 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
29 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
30 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
vote
1answer
36 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 ...
-3
votes
2answers
38 views

Number of 15 digit numbers [closed]

How to find the number of 15 digit numbers such That each digit is not less than the one directly succeeding it. eg:998775333211100
0
votes
1answer
43 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
vote
1answer
39 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
vote
1answer
18 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 ...
1
vote
0answers
10 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
43 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
votes
2answers
28 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
40 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
41 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
39 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
13 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
vote
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 ...
1
vote
2answers
23 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
33 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
50 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
49 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
40 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
93 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
vote
1answer
12 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
16 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 ...
0
votes
0answers
9 views

Sign Of Permutation That Is Written As C Different Cycles

prove: if $\sigma\in S_n$ is a factorization of $c$ disjoint cycles so $Sgn(\sigma)=(1)^{n-c}$ We know the one cycle sign is $(-1)^{l-1}$ so $c$ of them is $(-1)^{l-1}\cdot ...
1
vote
1answer
25 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
66 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
19 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
19 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
votes
2answers
38 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
vote
1answer
139 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
247 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
186 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
16 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
votes
0answers
23 views

Catch the fraud! [migrated]

Ok now this is one tough math question but fun to try. If this site does not tolerate such matter please let me know and I'll remove this. There are 10 gold smiths in the town and a rich businessman ...
0
votes
1answer
20 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 ...
-1
votes
1answer
28 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
24 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
39 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
64 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
votes
2answers
27 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 ...