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

learn more… | top users | synonyms (1)

-1
votes
2answers
16 views

Matrices for action wrt basis

Consider the permutation representation where $G=S_3$ on $\mathbb{C^3}$ with the action: $\pi(g)e_i=e_{g(i)}$ $W=\{ \lambda_i e_i ; \sum \lambda_i=0 \}$ is an invariant subsoace of vector space $V$ ...
0
votes
0answers
11 views

Representatives of the conjugacy classes of s5 [duplicate]

List the partitions of 5 and corresponding representatives of conjugacy classes in s5. What is the procedure to find the representatives of the conjugacy classes?
1
vote
1answer
33 views

Order of subgroup of symmetric group

Let $X$ be a finite set, i.e. that $|X| = n$, and let $G = \operatorname{Sym}(X)$ be the symmetric group on $X$. Let $Y \subseteq X$ be a subset of $X$ and define the subset $G_Y \subseteq G$ to be ...
5
votes
1answer
54 views

How many $5$ card poker hands contain at least $1$ red and $1$ black card?

How many $5$ card poker hands contain at least $1$ red and $1$ black card? I used inclusion-exclusion to calculate my answer. The number of total poker card hands are:$$52\choose 5$$I have $26$ red ...
0
votes
0answers
38 views

Is this a good way of generating unique permutations?

This is something that I thought of on my own, though I am sure that I am not the first to think of it. The easiest way to explain this is by using an example. Suppose we want to find the unique ...
4
votes
3answers
87 views

Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
5
votes
2answers
210 views

{0,1}-matrix and permutation matrices

A permutation matrix is a square matrix with exactly one $\textbf{1}$ in each row and column, and zeros in all other positions of the matrix. Let $M$ be an $n\times n$ $\{0,1\}$ matrix with exactly $m$...
0
votes
1answer
32 views

Bases of Vector Spaces and Permutations.

Could anyone let me know how I go about this question? I have no idea what to do with the permutations. I don't even understand what I'm being asked here. Thanks. Suppose that $(u_1,\dots,u_n)$ and $(...
2
votes
1answer
342 views

Count arrays with each array elements pairwise coprime

Given two integers $N$ and $M$ , How to find out number of arrays A of size N, such that : Each of the element in array, $1 ≤ A[i] ≤ M$ For each pair i, j ($1 ≤ i < j ≤ N$) $GCD(A[i], A[j]) = ...
0
votes
0answers
40 views

Doubt in Circular Permutation: 4 Americans and 4 English are seated on a round table.No Two americans sit together.Find the number of ways.

The question is 4 Americans and 4 English are seated on a round table.No Two Americans sit together.Find the number of ways. So,after this I did: $(4-1)!$ for seating the Americans around the table....
2
votes
0answers
74 views

Number of ways of selecting teams in a competition

We have $25$ countries and $100$ teams. Teams can have variable sizes. Each team consists of a combination of players from different countries. Now we have to select $13$ teams in total subjected to ...
1
vote
2answers
16 views

Cycle decomposition of an element of prime order

I am reading the proof of the theorem that every alternating group $A_n$ is simple for $n \ge 5$ in Artin's Algebra. In one step, Artin said We are given that $N$ is a normal subgroup and that it ...
-1
votes
1answer
31 views

Number of binary strings containing at least n 1's

I have 53 binary digits and I need to calculate how many combinations of 1's and 0's can be generated where there are at least 40 1's in the combination. How can this be calculated?
0
votes
3answers
38 views

What is the number of elements $x \in S_n$ such that the cycle containing $1$ in the cycle decomposition of $x$ has length $k$.

Let $S_n$ denote the group of permutations of $\{1,2,3, . . . , n\}$ and let$ k$ be an integer between $1$ and $n$. I need to find the number of elements $x \in S_n$ such that the cycle containing $1$ ...
1
vote
1answer
34 views

Rearranging Digits of a Number

How many different numbers can be obtained by rearranging the digits of 1,273,421,695? Would it be C(10,2)*C(10,2)*P(8,6) = 40 million, 824 thousand Or would it be (10*10*8*8*6*5*4*3*2*1)/(2!*2!) = ...
0
votes
1answer
31 views

Formal way to express the number of lists of $k$ objects from $n$, having $i$ unique elements

Say that I have a matrix of the $n^k$ ordered lists of $k$ objects from a supply of $n$, with replacement (which I am not quite sure how it's called). Note that $k$ may be greater, equal, or less than ...
0
votes
3answers
37 views

Sum of all distinct numbers made

Question: Find the sum of all distinct four digit numbers that can be formed using the digits 1; 2; 3; 4; and 5, each digit appearing at most once. I have no clue as to where to begin this question. ...
1
vote
1answer
31 views

Expected sum value of permutaion

We have a set(A) of N elements. Let's assume elements are e1,e2,e3..etc. Value of each element can be 0 or 1. Another set of N elements(set B) are given, ...
2
votes
1answer
34 views

For counting permutations with identical objects, why does dividing nPr by the factorial of the number of identical objects give the correct answer?

I can find plenty of sites that say that this works, but I can't seem to find an explanation for why it works. I'm rather stumped.
0
votes
1answer
29 views

Linear Algebra - Permutations

Is it possible to multiply two permutations of different lengths together? If so how would you go about doing it?
1
vote
1answer
63 views

How many ways to arrange these gifts? (Inclusion-exclusion\derangement)

Each one of 30 people has bought 2 identical presents for the poor (every person's gifts are different from everyone else's). All the gifts were put in a large bag. In turns, 30 poor people ...
2
votes
1answer
35 views

Using the Binomial Identity, prove that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Using the Binomial Identity, prove that: $${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$$Because this is in the form of a Binomial Coefficient, I can break down the LHS further:$$\left(...
1
vote
2answers
47 views

Suppose a coin in tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences…

Suppose a coin is tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences are there in which there are at least $5$ tails in a row? I know this is Permutation with repetition. My ...
0
votes
1answer
30 views

How does $9\choose 4,3,2$ $=8$ $7\choose 4$

Can someone please explain to me how $9\choose 4,3,2$$=8$$7\choose 4$? From my understanding $9\choose 4,3,2$$ = $$9\choose 4$$5\choose 3$$2\choose 2$$=$$9\choose 4$$5\choose 3$$\cdot 1$ But for ...
4
votes
1answer
29 views

Counting permutations in $S_n$ with $1,2,..,k$ all in same cycle

The number of permutations in $S_n$ for which the first $k$ items $1,2,...,k$ are all in the same cycle can be shown (by a somewhat tedious argument) to be $n!/k.$ I'm looking for less computational ...
1
vote
0answers
34 views

Finding a particular permutation

Simple Notation: For a permutation $P=(a_1,a_2,...,a_n)$ , we define $\{P_k\} = \{a_1,a_2,..,a_k\}$. (i.e. set of first $k$ numbers). Problem: Given $N=\{1,2,3,..,n\}$ and $m$ subsets of it, $S_1, ...
3
votes
0answers
21 views

Determinant of $\delta$ function

Let $$\delta_i^j=\left\{ \begin{aligned} 1 ~~~~~~i=j \\ 0 ~~~~~~i\ne j \end{aligned} \right. $$ $1\le i,j\le n$. How to prove $$ \begin{vmatrix} \delta_{j_1}^{i_1} ~...~ \delta_{j_n}^{i_1} \\ \\ \...
1
vote
0answers
14 views

Calculating the number of permutations that do not have at least one set of duplicate elements adjacent.

Ok, so I've got a set of elements, some are duplicates but each are considered unique as far as set-making goes. I need to find how many permutations exist that do not put two of the duplicates next ...
1
vote
1answer
19 views

Permutations of $n$ objects where $r = n -1$

In my text book the question is as follows: Find the way in which $5$ persons can sit in a row if two insist on sitting next to each other. They give the answer as $48$. I fail to understand how ...
0
votes
1answer
36 views

Mr and Mrs Zimmerman want to give their baby a first name and a second name so that the baby's three initials are in alphabetical order.

Mr and Mrs Zimmerman want to give their baby a first name and a second name so that the baby's three initials are in alphabetical order. How many different initials could this baby end up with eg. B,G,...
1
vote
2answers
31 views

Permutations and Combinations based probblem

Find the value of the expression: $$ 1+1\times1!+2\times2!+3\times3!+.....+n\times n! $$ It is a problem based on the concept of permutations and combinations I don't have a perfect idea to solve ...
12
votes
0answers
106 views

Covering pairs with permutations

Consider an $n \times n$ matrix $M_n$ with the following properties: Each row is a permutation of $A_n \equiv \{1, 2, ..., n\}$. Every ordered pair $(i,j)$, $i,j \in A_n$, $i \neq j$, appears as a ...
-2
votes
1answer
17 views

For any permutation $ \sigma \in S_n$, $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even when $n$ is odd [closed]

Let σ be a permutation of ${1, 2, 3, . . . , n}$, n odd. I want to show that $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even. Thank you.
-2
votes
1answer
18 views

Numbers of words allowing repetition [duplicate]

10 different letters of an alphabet are given. words with 5 letters are formed from these given letters.I have to determine the number of words which have at least one letter repeated. Answer is - ...
0
votes
2answers
29 views

sum of numbers formed by permutations

I have digits 2,3,4,5. I have been asked to find the sum of all 4 digits the numbers that can be formed using these digits without repetition such that all are included in the number. Can someone ...
1
vote
1answer
31 views

I need help answering a few simple math problems related to permutations and probability

Question 1: How many words can you make from the letters Texas if repeats are not allowed? Question 2: How many words can you make from the letters Texas if repeats are allowed? Question 3: What is ...
0
votes
1answer
43 views

Combinatorics problems involving permutations

Let $A= \{ 1,2,3,...,n\}$ a set and $f:A \to A$ a permutation of the set A. We call a number $x \in \{ 2,3,...,n-1 \}$ special if $f(x)>\max \{f(x-1),f(x+1) \}$ or $f(x)<\min \{f(x-1),f(x+1) \}.$...
-1
votes
1answer
22 views

How many possible placements are there for a Battleship puzzle?

I am studying the NP-Completeness of the battleship puzzle; the pencil and paper game found in newspapers and not the more popular 2-player version. I understand why the puzzle is NP-Complete because ...
0
votes
1answer
32 views

Find commuting elements within a permutation group

The question is like this: IF $G=S_5$ and $g=(1\quad 2\quad 3)$, determine the number of elements in $H=\{x\in G:xg=gx\}$. To do the question, first it says $$x(4)=(x(1\quad 2\quad 3))(4)=(1\quad 2\...
1
vote
2answers
21 views

Sign of composition of transpositions

Let $\sigma \in S_n$. Definition: Suppose that $\text{sign}\sigma=(-1)^N$, where $N$ - number of inversions in permutation $\sigma$. Suppose that $\tau_1$ and $\tau_2$ transpositions. How to prove ...
1
vote
1answer
23 views

Sign of permutation. Confusing example

Let $\sigma=(2314)\in S_4$. We have different definitions of sign of permutation. 1) Our $\sigma=(24)(21)(23)$ hence $\text{sgn}\sigma=(-1)^3=-1.$ 2) Our $\sigma$ has two inversions namely $(2,1)$ ...
0
votes
1answer
19 views

Does every partition of n correspond to some permutation of [1,2, … n]?

It is known that every permutation can be decomposed into disjoint cycles. The cycle type gives the length of each cycle. The sum of cycles length is n. I am wondering whether every partition of n ...
1
vote
1answer
36 views

Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...
2
votes
1answer
53 views

what is the smallest non-abelian finite group which has normal, non-abelian subgroups (plural)

I am looking for smallest example of a group $G$ such that: $G$ is a finite, non-abelian group $G$ is not simple $G$ has non-trivial, proper, normal subgroups: $H_1, H_2, \dots $ $H_1, H_2, \dots $ ...
0
votes
0answers
23 views

Cycle structure of the generators of the dihedral group

Would the following be correct about generating the dihedral group $D_n$ by permutations? If $n$ is even, the group can be generated as $\langle(2\quad n)(3 \quad n-1) \ldots (\frac{n}{2}-1 \quad \...
0
votes
1answer
23 views

permutations of n objects

Does the number of permutations of $n$ objects, $r$ alike of one kind and $n−r$ alike of another kind, always equal the combinations of n different objects taken $r$ at a time? Explain. I know ...
1
vote
2answers
52 views

Partition of natural number not equal to factorial

I wish to prove the following statement so I can use it as a lemma for a group theory result. To be honest I have not tried much yet, my intuition tells me this is going to be connected to the ...
0
votes
1answer
21 views

Arrangements of crew in two sides of a boat - permutations and combinations

A boat crew consist of 8 men, 3 of whom can row only on one side and 2 only on the other. The number of ways in which the crew can be arranged is This is a problem my math teacher has given to ...
-3
votes
1answer
75 views

Simple Question on Binomial theorems… [closed]

I have tried to solve this question by putting the value of each coefficients but it is really becoming very lengthy.... what i got was this number 10510100501... But how to get this in the required ...
7
votes
2answers
555 views

Number of ways of visiting N places

A tourist wants to visit $N$ cities, each numbered from $1$ to $N$, but he wants to visit them in a weird order. A weird order is such in which no city numbered $i$ is the $i$-th to visit in his ...