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

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multiplication of permutation

I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, $a=(1\ 3\ 5\ 2)$, $b=(2\ 5\ 6)$, $c=(1\ 6\ 3\ 4)$, $ab=(1\ 3\ 5\ 6)$, ...
3
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1answer
70 views

If matrix $A$ is invertible, then there is a permutation of its rows leaving no-zeros on the diagonal

I need to prove this statement: "If $A$ invertible, then exist a permutation of its rows leaving no-zeros on the diagonal" and I tried using the definitos of invertible matrices and $LU$ ...
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2answers
37 views

Need help in confirming the answer to a combinatorics question?

I need help to confirm my answer for the following question "There is an alphabet of size 40 and this alphabet is used for forming messages in a communication system. If 10 of these alphabets can be ...
2
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1answer
45 views

Number of ways to order items

How many ways are there to put 10 red and 9 blue balls in a sequence so that for every index the number of red balls up to and including this ball is greater than the number of blue balls? It means ...
3
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1answer
97 views

Is this notation $\prod\limits_{k=k}^n k $ valid for expressing this product? (ways of arranging $k$ things in $n$ places)

I want to express how many ways you can arrange $k$ things in $n$ places. $$\prod\limits_{k=k}^n k = k (k+1) (k+2)\cdots(n-1) n$$ Edit (added) { The example from which I started thinking about this ...
3
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2answers
39 views

No. of ways to arrange 4R and 3L so that there is exactly 4 times change from L to R or R to L.

I have to arrange 4R and 3L in such a way to know number of times there is 4 changes in the alphabet of sequence. For instance consider the sequence RLRLLRR, this sequence has 4 places where R and L ...
0
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1answer
42 views

Predict number of Birthdays for 1000 person of same class in next 365 Days

I want to know an approximate number of birthdays for a class where each month 1000 Persons are added up. Like 1st month its 1000, 2nd month its 2000, 3rd month it is 3000 And so on. Now lets say ...
0
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1answer
85 views

Partition numbers with restriction on the greatest part *and* on the number of positive parts

I’m looking at partition numbers. OEIS A008284 says that the number of partitions of $n$ in which the greatest part is $k$, $1 \le k \le n$, is equal to the number of partitions of $n$ into $k$ ...
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0answers
41 views

For $k$ random perms of an $n$-set $\mathrm{Pr}[\sigma_1\cdots\sigma_k=\sigma_k\cdots\sigma_1]\xrightarrow{k\rightarrow\infty}\frac{2}{n!}$?

Q. Fix $n \geq 2$, and choose $k$ random permutations $\sigma_1\sigma_2\cdots\sigma_k \in S_n$ uniformly at random. Is true that ...
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2answers
219 views

Probability of $\alpha\beta\gamma=\gamma\beta\alpha$ for random permutations of a finite set?

Following up on my previous question Probability that two random permutations of an $n$-set commute?, here's a related question for three elements. Q: If $\alpha,\beta,\gamma$ are chosen uniformly ...
0
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1answer
85 views

How many nonnegative integer matrices of size $N$ have all row and column sums equal to $D$?

Given the positive integer $N$ and $D$, generate all the non-negative integer matrices which satisfy matrix dimension is $N\times N$; sum of each row elements equals to $D$ sum of each column ...
1
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1answer
228 views

Number of unique permutations of a 3x3x3 cube

Given a 3x3x3 cube (like a rubik's cube) where each of the 27 cubes has a distinct number, how many unique permutations are possible? Simple rotations of the entire cube should not be counted. The ...
3
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1answer
165 views

Possible permutations of a 3x4 cube puzzle

My kids have this 3x4 cube puzzle, you know, the one where a picture is formed if you assemble the cubes correctly. In reality you can create 6 different pictures, using different sides of the cubes. ...
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2answers
45 views

Combination problem technique

How many numbers can be formed from 1, 2, 3, 4, 5 (without repetition), when the digit at the units place must be greater than that in the tenth place? It can be easily solved that, the total ...
4
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2answers
116 views

Probability that two random permutations of an $n$-set commute?

From this MathOverflow question: It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. -- Benjamin ...
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1answer
40 views

Factorial and combinations question.

Any help with these would be greatly appreciated... 1) How many arrangements are there of the letters of the word SAUSAGES ? if the A’s must be together and the S’s apart? (answer apparently 240 ...
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0answers
51 views

Mapping permutations with repetitions to an index

I am looking for a way to map permutation with repetitions written in big array (for example with million elemenths) to an index written in other array. Every element in array can have values from 0 ...
0
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1answer
45 views

Simple Permutations/Combinations Question

A group of 5 men and 5 women stand in line to have their photo taken. How many ways can they stand in line if no two men and no two women stand together? My method: _M_M_M_M_M_ Male * Female = 5P5 ...
0
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2answers
84 views

Number of permutations without repetition without having to pick all elements

I am looking for a formula to calculate the number or possible permutations when: a) repetition is not allowed and b) you don't have to pick all elements from the pool. So I got n elements and I want ...
2
votes
2answers
60 views

Commutative permutations

Observation: Two disjoint permutations are commutative. For e.g $\ (1357)\in S_8$ and $\ (2468)\in S_8$. Formulate and prove(if possible) a Theorem generalizing your observation about commutative ...
0
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1answer
37 views

Completely unique set in permutation

I have tried searching online for the answer and can't quite get one for my specific problem. My use of terminology is probably not helping (I don't study math). I think I know the answer but would ...
0
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1answer
75 views

Abstract Algebra:Permutations

1)Express the permutations $\alpha=(24)\in S_4$ and $\beta=(1)\in S_5$, as sets. a) Describe the permutations which are reflexive b) What types of permutations are partial orders. Attempt of a ...
4
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1answer
114 views

Permutations: If I know $\alpha$ and the cycle structure of $\alpha\beta$, can I find $\gamma$ for which $\gamma\beta$ also has this cycle structure?

Suppose we have two permutations $\alpha$ and $\beta$ (of a set $S$ of size $|S|=n$), and I know $\alpha$ and the cycle structure of $\alpha\beta$. But I don't know $\beta$. Can I find a ...
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4answers
159 views

Permutation, Combinatorics

Stuck here : there are 100 objects labeled 1, 2,...100. They are arranged in all possible ways. How many arrangements are there in which object 28 comes before object 29. My approach : Consider ...
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0answers
48 views

identifying a subgroup of $S_8$ generated by 4-cycles

Let $G \subseteq S_8$ be the subgroup generated by some 4-cycles. If we number the elements $1,2,\dots, 8$, the 4-cycles are $(1234),(5678),(1485),(2376),(1265),(4378)$ I am not sure if I have ...
0
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0answers
56 views

Number of 3 digit numbers that can't be confused if they are rotated

NOTE: Here confusion refers to the fact that the number if written on a piece of paper would appear as different number if viewed upside down. 18 may be viewed as 81 and so on Can anyone find a ...
3
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2answers
78 views

Working with groups. Finding the inverse of some $S_9$

I want to compute the inverse of: $\begin{pmatrix} 1&2&3&4&5&6&7&8&9\\3&2&1&6&5&9&4&8&7 \end{pmatrix}$ Sorry about alignment(they are ...
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1answer
82 views

Infinite set and Permutation group

Suppose $X$ a infinite set and $S_X$ is the permutation group of $X$. Prove that any proper subgroup of $S_X$ has infinite index.
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3answers
93 views

How do i find the order of a permutation in the group $S_n$

How can i define the order of a permutation without doing the permutation again and again? Example: say $σ=(1-->2,2-->3,3-->5,4-->1,5-->4)$ in $S_5$.
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2answers
83 views

4 bet selections with all possible doubles

I have selected 4 football matches to bet on this evening and noticed I have the option to automatically place a bet on every possible double going. Betfair states that with 4 selections, you can ...
2
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1answer
122 views

Blocks in permutation group theory (D&F)

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Let $G$ be a transitive permutation group on the finite set $A$. $A$ block is a nonempty subset $B$ of $A$ ...
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0answers
69 views

Relation between permutations and fourier transform?

i dont know if this is already addressed somewhere (searching around did not find sth). The motivation is to find a way to generate or produce permutations using concepts from continuous mathematics ...
0
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1answer
50 views

Conjugating a permutation

I am trying to see that two permutations are conjugate exactly when they have the same cycle decomposition. I fail to see that $$r(i_1,i_2,\dots,i_k)r^{−1}=(r(i_1),r(i_2),\dots,r(i_k))$$ Any thoughts ...
2
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1answer
61 views

Group of permutations and disjoint cycles.

Two cycle $[i_1,..., i_r]$ and $[j_1,...,j_s]$ are said disjoint if no integer $i_\eta$ is equal to any integer $j_{\mu}$. Prove that a pemutation is equal to a product of disjoint cycles. I was ...
0
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4answers
196 views

Total number of unique permutations in a lexicographic set

Given N digits that can form a set, how many total unique permutations of the set can be generated if we do not care about the order of the digits in the set? Specifically, I'd like to ask for ...
6
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2answers
153 views

Are all Sylow 2-subgroups in $S_4$ isomorphic to $D_4$?

I was assigned to show that every Sylow 2-subgroups in $S_4$ is isomorphic to $D_4$. So I figured, since $|S_4|=24=2^3\cdot 3$, every Sylow 2-subgroup has either the form: $\langle ...
4
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2answers
436 views

Finding the centralizer of a permutation

I need to find the centralizer of the permutation $\sigma=(1 2 3 ... n)\in S_n$. I know that: $C_{S_n}(\sigma)=\left\{\tau \in S_n|\text{ } \tau\sigma\tau^{-1}=\sigma\right\}$ In other words, that the ...
4
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2answers
163 views

Count Number of Sequences

The question is: Given a sequence of positive integers A={1,2,3,...,N}. Count the number of sequences you can get after making K swaps between adjacent element on it for a given N ? My approach: My ...
2
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3answers
68 views

Why is the number of arrangements of $n$ objects equal to $n!$?

I hope this question does not seem vague, but whats the logic/reason behind that e.g. One can arrange 5 colours in $5!$ different ways if all of the colours are different?
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0answers
313 views

Count swap permutations

Given an array A = [1, 2, 3, ..., n]: ...
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3answers
38 views

DNA Sequence Distinct Way

we know The genetic code is based on the four nucleotides adenine (A), cytosine (C), guanine (G), and thymine (T). These are connected in long strings to form DNA molecule. with three A, one C, two G ...
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1answer
74 views

Example of non-Abelianness of symmetric group for graphs

I know that for $n \ge 3$, $S_n$ is non-Abelian. I would like to work out an example in terms of graphs so to make it sure that I understand it right. A symmetric group of graphs of four vertices, ...
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1answer
28 views

A problem on counting

Consider a number system which does not have the number 7 but has all other numbers. So the numbers are $1,2,3,4,5,6,8,9,10,11,12,13,14,15,16,18,...$. I want to find what is the $10^k$ number, where ...
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2answers
176 views

how many ways 3 pairs can be selected out of 6 students?

Out of "6 students" how many ways can 3 pairs be selected for assigning homework ?
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120 views

Two 3-cycles generate $A_5$

I want to solve the following exercise, from Dummit & Foote's Abstract Algebra Let $x$ and $y$ be distinct 3-cycles in $S_5$ with $x \neq y^{-1} $. Prove that if $x$ and $y$ do not fix a ...
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1answer
25 views

Finding String permuatations

Given a string of characters(a-z) ,find subsequence such that every character is strictly greater than all previous characters in that subsequence. Example if S=abc then there are 7 subsequences ...
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1answer
198 views

Largest symmetric group contained in alternating group

I know that for $n \geq 3$, the alternating group $A_n$ contains a subgroup which is isomorphic to $S_{n-2}$, namely $$\langle \{(i \;i+1)(n-1 \;n):1 \leq i \leq n-3\} \rangle.$$ I was wondering what ...
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1answer
47 views

Prove summation related to cycles

Let $b_r(n,k)$ be the number of n-permutations with $k$ cycles, in which numbers $1,2,\dots,r$ are in one cycle. Prove that for $n \geq r $ there is: $$ \sum_{k=1}^{n} ...
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2answers
48 views

Fundamental confusion on set theory and permutations

I am confused on the following: A set does not have any order. Now I read that a permutation is a bijection of a set. But doesn't this imply an order? I mean a bijection is a one-to-one function from ...
0
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1answer
52 views

Maximum winner matches

N players take part in tennis championship. In every match loser is out. Two players can play a game if in that moment the difference of played games of that two players is not more than 1. They are ...