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

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Permutation Group-$S_{10}$

How many elements of order $30$ are there in the symmetric group $S_{10}$? I worked out and got $10500$ - using Computations and adjusting each individual cycle's position ( $2-3-5$ , $3-2-5$ etc). ...
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213 views

Circular permutations - $n$ sitting at a round table without repeating neighbors

I hope this isn't a duplicate - the problem is to find the number of ways of sitting $n$ people (who initially were sitting in the order $1, 2, \dots,n$, with $1$ and $n$ being neighbors) at a round ...
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76 views

Combinatorics question - How many different ways to change sitting order

Six children ($a$ through $f$) are playing on a carousel with 6 seats such that $a$ is sitting in front of $d$, $b$ is sitting in front of $e$, and $c$ is sitting in front of $f$. How many ways are ...
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1answer
25 views

What's the right way to look at this permutations problem?

I'm having trouble with this problem. It seems very simple. Here it is exactly: Obtain the number of three–letter permutations possible for the group of letters shown. S, E, V, E, N ...
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1answer
68 views

How many functions are not one-to-one?

I am having some trouble starting this question. If we have two sets. Set A of size m where m≥1 and set B of size n where n≥1. How many of the functions f : A→B are not one-to-one? I know that the ...
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1answer
58 views

How to show that $A_n$ generated by all permutations of the form (ab)(cd)?

I need to show that if n $\ge$ 5 then $A_n$ generated by all permutations of the form (ab)(cd) (a b c d are all different) Can I use the fact that union of conjugacy classes is normal ? Its clear ...
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1answer
58 views

Prove a polynomial in Fq is a permutation polynomial of Fqn with a necessary and sufficient condition

P.S This is the best Math Expression I can edit. I am real shameful, where can I find the introduction of typing in this webset? thank you Exercise7.13 Let\[f\left( x \right) = \sum\limits_{i = ...
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1answer
98 views

How many matrix colorings are possible?

We have a small square matrix having size up to $8$. And we have a large number of colors up to $10^6$. In how many ways we can color the matrix so that all the same color cells are not adjacent? ...
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2answers
165 views

Find all the elements in $S_4$ that commutes with $(12)(34)$.

Find all the elements in $S_4$ that commutes with $(12)(34)$. And show the algorithm of the process.
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2answers
42 views

Ternary in a 10-digit string

I encounter an interesting question and can't seem to form a logic for solving it. We need to form a 10-digit string using 0, 1, or 2 (ternary string). There should be exactly 3 0's. In total, how ...
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1answer
39 views

Convex Hull of cyclic Permutations

It is known that the convex hull of permutation matrices yields exactly the stochastic matrices. I am interested in the convex hull of cyclic permutation matrices. Trivially this is a subset of the ...
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3answers
176 views

Generalization of permutation matrix

For integers $n$ and $k$, I am interested in $n\times n$ matrices with exactly $k$ non-zero entries in each row and each column. The case $k=1$ corresponds to (generalized) permutation matrices. In ...
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0answers
67 views

Probability/selection problem

Assume that we have $N$ items of $M$ distinct types in a closed bag. We also have $K$ bowls $(K \leq M)$ that can hold only items of same type. In the beginning bowls are empty. And bowls can hold a ...
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2answers
96 views

Combination on jury selection

$20$ women (including Alice and Betty) and $12$ men show up for jury duty. In how many ways can you select a jury of at least $5$ women and at least $5$ men if one of Alice or Betty must be selected, ...
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177 views

Functoriality of the correspondence between oligomorphic actions and $\aleph_0$-categorical theories

If a group $G$ acts on a set $X$, then the action is said to be oligomorphic if the number of orbits of $X^n$ under the action is finite for each $n$. There is a classic theorem in model theory that ...
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1answer
91 views

A question about permutation and cyclic group

Let $S_4$ be the group of permutations on $\{1,2,3,4\}$ and let $G = S_4\oplus \Bbb Z_4$. Find the order of the largest cyclic subgroup of $G$.
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90 views

Permutations for paths

What is the counting sequence for paths from $$(0,0)\text{ to }(n,n)$$ where $$2n$$ is the size of the path (number of steps) and n can vary over all nonnegative integers. I don't know how to ...
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1answer
116 views

Four Children Combination Problem

This is a problem that has haunted me for some time There is two family. First Family : ...
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3answers
648 views

Finding the probability that three friends get into the same group [closed]

I am stuck with the following problem: Students of a school are divided into $\,4\,$ groups. What is the probability that three friends get into the same group ? The options are : ...
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1answer
46 views

Number of orders and combinations

I have just done these two questions and I have answers for them but I am not sure if they are correct. A jazz band is to give one concert in each of nine selected cities. Calculate the total ...
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2answers
78 views

Permutation question

I'm trying to solve this question In a photo there are three families six Greens, four Browns and seven Grays arranged in a row. The Browns have had an argument so no Brown will stand next to another ...
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8answers
7k views

Calculating the number of possible paths through some squares

I'm prepping for the GRE. Would appreciate if someone could explain the right way to solve this problem. It seems simple to me but the site where I found this problem says I'm wrong but doesn't ...
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1answer
124 views

Finding the number of different relations and functions

This must be a very stupid question. Let set $A=\lbrace{a,b\rbrace}$ and $B=\lbrace{1,2,3\rbrace}$. The total number of relations from $A$ to $B$ is $6$. We can calculate this as a has $3$ choices and ...
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1answer
61 views

How to convert a permutation to permutation polynomial?

Let Fq be the finite field with q elements, where q is a prime power. A permutation on Fq is a bijection from Fq to itself. Let Fq[x] be the ring of polynomials in a single indeterminate x over Fq. A ...
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3answers
180 views

Example of a simple graph isomorphic to a permutation group.

I'm taking a first course in graph theory this semester and I'm working trough Graph Theory with Applications by J.A. Bonday and U.S.R. Murty. I can't find an answer to question 1.2.12(f): (a) ...
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1answer
82 views

Taking square root of cycle permutations

Let $\alpha$ and $\beta$ be permutation cycles of $\{1,2,\ldots,n\}$ such that $\alpha^2=\beta^2$ Can we conclude that $\alpha=\beta$, if (a) $\alpha,\beta$ are odd? (b) $\alpha,\beta$ ...
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1answer
111 views

Permutation that fix elements within set is subgroup?

(a) Let $A$ be a finite set, and $B\subseteq A$. Let $G$ be the subset of $S_A$ (permutations of $A$) consisting of all the permutations $f$ of $A$ such that $f(x)\in B$ for every $x\in B$. Prove ...
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1answer
21 views

$11$ matches are to be played,Each having $3$ distinct outcome,

$11$ matches are to be played,Each having $3$ distinct outcome, in how many ways one can predict the outcomes such that $6$ outcomes turn out to be correct? My thought $11C_{6}\times 3^5$ am I ...
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2answers
450 views

What is the parity of permutation in the 15 puzzle?

You might know the 15 puzzle: $\hskip1.4in$ Concerning the solvability, Wiki says: The invariant is the parity of the permutation of all 16 squares plus the parity of the taxicab distance ...
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1answer
118 views

Intuition - Identities with 2-Cycles and 3-Cycles - Mulholland p. 69, 86 - Fraleigh p. 90

Jamie Mulholland p. 69 Theorem 6.1 or Fraleigh p. 90 Corollary 9.12 Any permutation of a finite set of at least two elements is a product of 2-cycles. $1. (a_1, a_2, ···,a_n)= (a_1, a_n)(a_1, ...
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137 views

permutation and combination confusion

Can you point out where am I going wrong ? I wanted to select 5 men from 7 men . This can be done very easily as 7C5 ways = 21ways, but I was confused as to why the following calculation didn't work ...
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1answer
119 views

How many combinations does Android pattern have?

Rules- 1) At-least 4 and at-max 9 dots must be connected. 2) There can be no jumps 3) Once a dot is crossed, you can jump over it.
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1answer
40 views

Permutations from a set

Part of my problem is I can't figure out which question answers my problem. I'm not so familiar with the kind of math lingo that I know how to ask this question, so I'm gonna bumble my way through ...
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2answers
1k views

How do I find the maximum number of knights on a chess board?

I came across this problem and after thinking a lot I could not get any idea how to calculate it. Please suggest to me the right way to calculate it. Given a position where a knight is placed on ...
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1answer
57 views

How to arrange $n$ pairs of numbers so that this expression is minimized

Consider $n$ pairs of positive integers, $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$. Make a permutation $(a_1, b_1), (a_2, b_2), \dots, (a_n, b_n)$ of these pairs, such that for all $x_i, y_i$, a ...
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0answers
175 views

Count the number of restricted multigraphs

Suppose I have a multigraph with the following set of restrictions: every vertex can have up to $c$ edges two vertices can be connected by a maximum of $c-1$ edges loops may or may not be allowed ...
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132 views

Wikipedia's Cayley Table and Pictures for 3 by 3 Permutation Matrices

Are there any explanations or clarifications of the pictures at https://en.wikipedia.org/wiki/Permutation_matrix#Permutation_of_rows_and_columns? ...
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172 views

How many $n$ digit numbers can be written using only certain digits?

For example, how would I calculate how many $5$ digit numbers I can write using only digits $0, 2, 2, 3, 3$? Is $4\times5\times5\times5\times5$ correct or am I missing something?
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1answer
62 views

Number of possible interleavings of two strings of lengths m and n

Given two strings, str1 = "AB", str2 = "CD" The interleavings are ABCD ACBD ACDB CABD CADB CDAB, I'd like to know if there is a general formula I can come up with in terms ...
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1answer
116 views

A canonical form for this equivalence relation on matrices

This question is inspired by http://cs.stackexchange.com/q/19250/755. Define the equivalence relation $\sim$ as follows: If $M,N$ are two $8\times 8$ (or $n\times n$ if you prefer generality) $(0,1)$ ...
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2answers
257 views

How many elements of order $k$ are in $S_n$?

I need to find how many elements of order $k$ are in $S_n$ (where $k \leq n$). So if $k$ is prime, it's easy: $k$ can't be the $\mathrm{lcm}$ of any integers besides itself and one's (which we're ...
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0answers
64 views

A few ways to transposition or cycle decomposition of permutation.

I've found exercise with permutation and resolve that but I'm not sure that my resolve is proper. Proof that every permutation $\sigma\in S_n$ can be decompose to product of transposition ...
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2answers
48 views

Find solutions of this equation

If $a+b+c+d = 30$ and $a,b,c,d$ lie between $0$ and $9$. How to find number of solutions of this equation.
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1answer
36 views

Is the approach to proving this expression for an n-choose-k algorithm correct?

I randomly encountered this post here, asking why this is true: ...
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108 views

Yet another Seating Arrangement Challenge

Not sure if this is the right place to ask this, but here goes (fingers crossed that someone can point me in the right direction): I am planning a singles "speed dating" type event, but it's a little ...
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1answer
154 views

Use of the commutator to reduce the degree of a permutation

I've found the following claim here (page 4, in the proof of thm 1.1). Set $A$ the support of a permutation $\tau$ in $S_n$, with $\deg\tau=|A|=k$. Given a permutation $\pi$ in $S_n$ such that ...
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1answer
50 views

find all subcartesian products of $S_4$ and $D_{12}$

The following exercise is from [Cameron, Permutation Groups]: Find all permutation groups of degree 10 which have orbits of length 4 and 6 and act on these orbits as the symmetric group $S_4$ and ...
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1answer
81 views

Does the isomorphic image of a stabilizer subgroup fix a point

Here is an exercise from [Birkhoff and MacLane, A Survey of Modern Algebra]: Let $\phi: G \rightarrow G'$ be an isomorphism between two groups of permutations. Let $S$ consist of those ...
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55 views

How many outcomes can this selection have?

Say I am setting up a timetable. One subject has multiple lecture groups, and all the lecture groups share a uniform amount of lectures. To further elaborate on the example, the subject I'm plotting ...
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1answer
261 views

Abelian subgroup in the symmetric group

Let $p$ be a prime number. Show that there is an abelian subgroup $P$ of order $p^p$ in $S_{p^2}$ such that every element in $S_{p^2}$ that isn't in $P$ does not commute with every element in $P$. ...