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

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Permutation test for equality between the distribution of $g$ population

I have a data matrix of 42000 observation and 12 variables I suppose to observe 12 samples of size $n_j$ from 12 indipendent random variables $Y_j,j=1,...,g$ I want do a permutation test for $$H_0: ...
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0answers
35 views

12 numbered pigeonholes and balls [duplicate]

This problem was inspired by this James Randi challenge. Given $12$ numbered ($1$ to $12$) pigeonholes and $12$ numbered balls (also from $1$ to $12$); what is the probability that a random ...
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1answer
18 views

Total number of integral solutions to the factors of a given numbet

Let $a$ be a factor of $120$ then what are the total number of positive integral solutions to $xyz=a$ including 120. The answer is $320$ . After wasting almost $15$ mins in getting the factors of each ...
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0answers
24 views

Number of ways to allocate balls

We are given $N$ buckets and $X$ ways to allocate balls in each possible pair of buckets. How many ways of distributing balls in all buckets exist ? For example - If we have $3$ buckets and $7$ ways ...
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1answer
25 views

Character of a representation on $S_3$ and irreducible representations

Here is the character table of S3: Consider $V=\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ with basis $e_{ijk} := e_i \otimes e_j \otimes e_k $ Let $\pi$ be the representation of ...
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2answers
49 views

How to calculate permutation $(12)^{-1}(12345)(12)$ [closed]

I was wondering if someone could help me find $(12)^{-1}(12345)(12)$ I need to know this for calculating conjucacy classes and then a character table, thanks
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0answers
24 views

Algorithmic complexity of testing whether a permutation belongs to a subgroup generated by a set of permutations

Let $S=\{S_1,S_2,S_3,\ldots,S_m\}$ be a set of permutations on $n$ symbols (in other words $S$ is a subset of a symmetric group on $n$ symbols) and $P$ be a permutation on $n$ symbols. What is the ...
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0answers
22 views

What happened to the “permutation-groups” tag? [migrated]

There used to be a "permutations-groups" tag, which I don't see anymore. What happened to it? Can it be put back?
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2answers
37 views

What does this definition of permutation mean?

A simple question. They give the definition of permutation as "a one to one mapping of the set onto the set of positive integers $\{1, 2,3,4, \ldots n\}$." What does this definition exactly ...
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0answers
13 views

K-permutation of given multiset

Is there a formula (or an algorithm) to calculate K-permutation of given multiset? The particular example: given a multiset of 15 balls in which every 3 are marked with one of the letters A, B, C, ...
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1answer
27 views

Permutations acting on coordinates of codewords

Let $\mathcal{C}$ be a binary code of length $n$. The automorphism group of $\mathcal{C}$ is defined to be the set of permutations in $S_n$ that take $\mathcal{C}$ to itself. The text by MacWilliams ...
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1answer
33 views

Permutation and Combination to find pairs

In how many different ways students can be paired such that no pair consists of 2 boys. Given :- Total students = 10, Girls = 7, boys = 3. What my approach is 3 boys can be paired with 7 girls like ...
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3answers
64 views

Number of permutations which are products of exactly two disjoint cycles. [duplicate]

Let $l_{n}$ denote the number of those permutations $f$ on the set $A=\{1,2,....,n\}$ such that $f$ is the product of exactly two disjoint cycles. Show that $l_{5}=50.$ I tried a lot but reached ...
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2answers
25 views

Understanding a proof about permutations from P.A.Grillet's “Abstract Algebra”

I need a hand in understanding the following proof of the following theorem(by P.A.Grillet in his textbook "Abstract Algebra"). Proposition $4.1$. Every permutation of $\{1,...,n \}$ is a product ...
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0answers
34 views

Find all permutation sequences with only neighbor swaps

I am trying to essentially do what was done at this link, except for five elements instead of four. The link gives all of the possible sequences of permutations of four elements with the following 2 ...
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3answers
63 views

Five people have applied for three different positions in a store. In how many ways can the positions be filled?

Five people have applied for three different positions in a store. If each person is qualified for each position, in how many ways can the positions be filled? Can someone tell me if I have to ...
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2answers
15 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 $V$ ...
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0answers
10 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?
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1answer
32 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
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1answer
51 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 ...
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0answers
36 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 ...
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3answers
84 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
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2answers
209 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 ...
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1answer
31 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 ...
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1answer
322 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]) = ...
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0answers
31 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 ...
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0answers
61 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 ...
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2answers
15 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 ...
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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?
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3answers
35 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 $1$ ...
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1answer
26 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!) = ...
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1answer
28 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 ...
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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. ...
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1answer
27 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, ...
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1answer
33 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.
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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?
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1answer
61 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
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1answer
31 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 ...
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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 ...
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1answer
29 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 ...
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1answer
28 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 ...
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0answers
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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, ...
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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} \\ \\ ...
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31 views

Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd S_\mathbb{N}$. [closed]

Let $S_\infty \subset S_\mathbb{N}$ be the set of permutations of $\mathbb{N}$ which are the identity on all but a finite number of elements. Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd ...
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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 ...
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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 ...
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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. ...
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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 ...
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95 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 ...
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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.