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

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1answer
13 views

Conjugation of permutation group $S_n$

I want to find the conjugacy classes of the permutation group $S_n$ To start with I think I have to prove that $\pi(\sigma_1\dots \sigma_m)\pi^{-1} = (\pi(\sigma_1)\dots \pi(\sigma_m))$. Where $\pi$ ...
1
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3answers
50 views

How to find a permutation $\sigma$ given the permutation $\sigma^2$?

How to solve the equation: $\sigma ^2 =\left({\begin{array}{*{20}c}1 & 2 & 3 & 4 & 5\\ 1 & 4 & 2 & 3 & 5\end{array}}\right)\ $ where $\sigma \in S_5$. Is there a ...
1
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5answers
512 views

In how many ways can $7^{13}$ be represented as product of $3$ natural numbers?

How i solved it: all possible non-distinct groups $(a,b,c)$ are, $a = 0 \Rightarrow (b,c) = (0,13)(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)$ $a = 1 \Rightarrow (b,c) = (1,11)(2,10)(3,9)(4,8)(5,7)(6,6)$ $a ...
1
vote
1answer
64 views

Permutation of numbers from multiple sets [May contain duplicate numbers among other sets], resulting in Non-Duplicate Set

We have 3 Data Sets. From each set we will be selecting few numbers. 3 from Set 1, 2 from Set 2, 3 from Set 3. Totally, we will get 8 Numbers from 3 Sets. The resulting sets shouldn't contain any ...
0
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1answer
20 views

number of combinations/permutations

if I have $n$ drawers and in each drawer I can only have 1 pen or 1 pencil for example if i have $3$ drawers the possible ...
0
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2answers
30 views

Probability of picking up one ball of each color

A box contains 6 red, 4 white and 5 black balls. A person draws 4 balls from the box at random. Let P be the probability that among the balls drawn there is at least one ball of each color. Find 455 * ...
0
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1answer
25 views

Possible number of sequences

How many finite sequences are there such that $x_{i} = 1$ or $ 2$ and $\sum_1^n x_{i} = 10$ ? Now I did it this way: number of $1$'s $\ $:$\ $ $0$ ,$2$ ,$4$ ,$6$ , $8$ , $10$ and ...
0
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3answers
51 views

Why is it true that $(a_1 \ a_2 \dots a_r) = (a_1 \ a_r)(a_1 \ a_{r-1})\dots(a_1 \ a_3)(a_1 \ a_2)$?

In the theory of permutation, a $r$-cycle $(a_1 a_2...a_r)$ is defined in the following way: Start from $a_i$, a permutation function $f$ sends $a_i$ to $a_{i+1}$. When $i=r, a_i \text{ will be ...
2
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1answer
20 views

Permutations of {1, 2 .. 30} where $a_n - a_n-m$ is divisible by m from {2, 3, 5}

There are $N$ permutations $(a_1,a_2,\dots,a_{30})$ of $1,2,\dots,30$ such that for $m\in\{2,3,5\}$, $m$ divides $a_{n+m}-a_n$ for all integers $n$ with $1\leq n <n+m\leq 30$. Find $N$. I really ...
0
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0answers
62 views

Any hint on : Every $A_{n}$ elemnt is $n$-cycles product.

[Added explanation] I found this exercise as follows in Hungerford : Abstract algebra (3rd edition) page 236, exercise number 40. Stated as follows : C.40. Prove that every element of $A_{n}$ is ...
0
votes
1answer
72 views

If $g$ is a permutation, then what does $g(12)$ mean?

In Martin Lieback's book 'A Concise Introduction to Pure Mathematics', he posts an exercise(page 177,Q5): Prove that exactly half of the $n!$ permutations in $S_n$ are even. (Hint: Show that ...
4
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3answers
532 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
-1
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1answer
20 views

How many structurally different latin squares of order 5 do exist?

I know the number of latin squares order 5 which start with 1 2 3 4 5 in the 1st row or column, that is 1344, but the greater part of that number consists of structural duplicates of each other. So, I ...
3
votes
1answer
36 views

Permutations of n objects taken r at a time ( r=1 to r=n ) where objects may be groups of same entities and it's sum

Given n objects where n1 objects are the same ,along with another group of n2 objects of same element etc.. such that Σni = n (i=1 to k). Assuming there are k groups of similar objects eg: in ...
3
votes
2answers
42 views

Permutation count of AABBC

Given a string: $AABBC=A^2B^2C^1$ I am trying to find the Total Permutations (this may be incorrect): $\dfrac{5!}{2!\cdot2!}=30$ My question is how would I find the partial sums (perhaps the ...
3
votes
2answers
37 views

How to conceptualize “dividing out” a number (e.g. in permutations, Bayes' Theorem)?

I'm trying to achieve a better conception of what it means to "divide out" a variable/number, because I'm currently have a lot of trouble justifying to myself why it actually works the way it does in ...
1
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2answers
57 views

How many two letter words can be formed from 26 English letters?

There are 26 English letters. From layman approach, How can one calculate the possible two letter words from these 26 English letters?
0
votes
2answers
32 views

Find the number of seven digit whole numbers in which only 2 and 3 are present as digits if no two 2's are consecutive in any number?

Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number? My Approach: We can make numbers and see like: ...
3
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2answers
35 views

Elements of $S_n$ which can not be product of $\leq n-2$ transpositions

It is well known that every element of $S_n$ can be written as a product of at most $n-1$ transpositions. This theorem is proved in all the books which discuss the permutation groups. But, I find that ...
0
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2answers
58 views

How many permutations of the word TOMORROW can be made if the O's can't be together?

I'm trying to answer this question. This is my attempt of solution: First we distiguish the O's and R's, then we have the word: $TO_1MO_2R_1R_2O_3W$. We have $8!-7!\cdot3!-6!\cdot 3!$ different ...
4
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2answers
89 views

Automorphism groups of vertex transitive graphs

Does there exist a finite nonoriented graph whose automorphism group is transitive but not generously transitive (that is, it is not true that each pair $(x,y)$ of vertices can be interchanged by some ...
4
votes
1answer
343 views

How many non-isomorphic permutation selections are on an arbitrary N x N square matrix with rotations applied?

My question is an extension to a classic one: On a square $N \times N$ grid, select exact $N$ cells that satisfy condition: only one cell selected in same row and column. How many solutions will ...
2
votes
2answers
78 views

Fill $8$ boxes with $60$ items

I have $8$ boxes and $60$ items: how many ways can I fill the boxes so that The order of the items in each box does not matter It does not matter which boxes are filled with which items. In other ...
0
votes
1answer
81 views

Permutation question in discrete mathematics. At least 1 out 3 members (P) from a total of 10 members

Im doing a question out of Discrete and Combinatorial mathematics by Grimaldi (4th Edition). I'm stuck on one of the questions and am trying to find an alternative way of doing it, that is not in the ...
3
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3answers
37 views

Combinatorics Question - Permutations and Supersets

I had a question that seems pretty straightforward, but I can't seem to wrap my mind around it. Let's say I have a bunch of elements in a set. {A, B, C, D, E}. How many permutations are there of ...
3
votes
2answers
61 views

Finding how many bits of length n there are

So we are starting on the section of combinatorics in my discrete math class and our instructor gave us a simple problem to see if we understood what we learned that day. The problem consists of three ...
1
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1answer
29 views

Functions that are “balanced” on the support of a permutation

Let $F = GF(2^n)$. Let $P(x), Q(x) \in F[x]$ be such that $P(x)$ is a permutation, while $Q(x)$ is not a permutation. For $\lambda \in F^*$ define the function $g_\lambda(x) = Tr(\lambda Q(x))$. Let ...
0
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2answers
360 views

How Can I calculate number of combinations/permutations with certain rules

Lets say I have 4 balls and when each ball is drawn it can be any value between 1-40 inclusive. If order isn't important then it would just be $40\cdot 39\cdot 38\cdot 37/4!$ But what if ball 1 had ...
0
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0answers
96 views

Permutation equivalence classes with kendall-tau distance

Consider a set $A=\{a_1,...,a_m\}\subset \{1,...,n\}$ for which $a_i<a_{i+1}$ for all $i = 1,\ldots,m-1$. Take any two distinct permutations $\sigma, \tau$ of $\{1,...,n\}$ such that $ ...
1
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2answers
54 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
2
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2answers
49 views

Permutations: Interpreting Image Notation

I have a problem in interpreting permutation. I think the definition and my interpretation of it don't match each other. Let $\sigma=(1\ 2\ 4\ 3)$, and $\tau=(1\ 3\ 2\ 4)$ in one-line notation. I ...
0
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2answers
37 views

Number of permutations of an integer

If $n$ is an integer, how many permutations are less than, equal to and greater than $n$? For example if $n=24335$, $43325\gt n$, $23345\lt n$, etc...
2
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4answers
97 views

When will Andrea arrive before Bert?

The question was as follows- on any given day, Andrea is equally likely to clock in at work any time from 8:50am to 9:06am. Similarly, Bert is equally likely to to clock in at work at any time ...
0
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2answers
34 views

Permutational Question

When I use the equation $P = \frac{n!}{(n - r)!}$ with n = 3 and r = 2, I get 6 permutations. Though, how do I get the amount of permutations without cross-duplicates(e.g A/B and B/A)?
4
votes
3answers
95 views

Why Composition and Dihedral Group have reverse order of operation?

NOTE - I didn't receive any answer in here and I think because my first post is not clear, so I entirely made another example: $K={\{id,r^2,r^4,s,r^2s,r^4s}\}$ is a proper subgroup of the dihedral ...
2
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0answers
251 views

permutation and combination advanced

I have n sets having values less than 100. I need to find how many arrangements could be made if I pick one element from each set such that in the given arrangement there are no duplicates? NOTE: A ...
1
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1answer
76 views

Intuitively and Mathematically Understanding the Order of Actions in Permutation GP vs in Dihereal GP

I define $r$ to be one rotation clockwise, and s to be reflection on the 'horizontal' line (see the figure). So I can make these bijections: (in clockwise order) $$\begin{align*} 1,2,3,4,5,6 ...
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2answers
76 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
2
votes
2answers
236 views

Number of 8 character passwords including numbers and letters without repetition

A password must be created with 8 characters. It can use number or letters, but they cannot be repeated (and letters are not case sensitive so we have only 36 characters). How many passwords are ...
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1answer
22 views

Probability of a user references in a network [closed]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
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: ...
2
votes
1answer
54 views

NP combination puzzle (Klotski)

I've written a C++ program to solve sliding puzzles games such as UnblockMe and Car Parking. I'm quite happy about it, since it solves various schemes in less than a second. Recently I fed the game ...
2
votes
1answer
92 views

Rank 3 permutation groups

Let $G \leq Sym(\Omega)$ be a finite permutation group of rank 3, $\alpha \in \Omega$ and $g,h \in G$ such that $x_1 := g(\alpha)$ and $x_2 := h(\alpha)$ are not equal. Now my question is: Is there ...
1
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0answers
25 views

How many permutations cover alternating/reverse alternating permutations?

Given integers $1$ through $2n$, let $S$ be set of ordering of integers that respect even alternating or reverse alternating permutations (https://en.wikipedia.org/wiki/Alternating_permutation) up to ...
2
votes
1answer
29 views

Where do I use that $G$ is a permuation group?

This is about question $4.1.7$ from Dummit and Foote, and also related to my previous question. The question is (summarised a bit): Let $G$ be a transitive permutation group on a finite set $A$. ...
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0answers
32 views

How to prove the theorem on group algebra of permutation group [closed]

How to prove the following theorem: If $t$ is a vector in group algebra of permutation group, then $\cal{y}t\cal{y}=\lambda_t \cal{y}$, where $\cal{}y$ is the Young operator of permutation group and ...
0
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1answer
44 views

Compose $(1243)$ and $(5)$

Checking my work. In either direction: $(1243)[1] = 2$ and $(5)[2] = 2$, so far we have $(1, 2,\ldots$ $(1243)[2] = 4$ and $(5)[4] = 4$, so far we have $(1, 2, 4,\ldots$ $(1243)[4] = 3$ and ...
0
votes
0answers
52 views

Finding permutation matrix

Let $P_{\pi}$ denote a permutation matrix associated to the permutation $\pi:\{1,...,n\}\rightarrow \{1,...,n\}$ and $\sigma$ denote the cyclic permutation $(1 2 ...n)$. If T is the $n\times n$ lower ...
0
votes
1answer
36 views

How many permutations $(a_i)_{i=1}^{30}$ of $\{1,\dots,30\}$ satisfy $m$ divides $a_{n+m}-a_n$ when $m \in \{2,3,5\}$ and $1 \le n<n+m \le 30$?

Define a permutation $(a_1,a_2,\dots,a_{30})$ of $\{1,2,\ldots,30\}$ as good if for all $m \in \{2,3,5\}$, we have that $m$ divides $a_{n+m}-a_n$ for all integers $n$ satisfying $1 \leq n < n+m ...
11
votes
2answers
249 views

Show that there is always a way to achieve det(A) > 0

a) Assume that $(a_1, ..., a_9)$ are different positive numbers. Let us make a 3x3 matrix $A_s$ by placing them arbitrarily into 9 positions available. Show that there is always a way to assemble ...