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

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Convention on Cauchy's two line notation for permutations

Let $n\in\mathbb{N}$. A permutation $\sigma\in S_n$ is denoted in Cauchy's two line notation as follow: \begin{pmatrix} 1 & 2 & \cdots & n \\ \sigma(1) & \sigma(2) & \cdots & \...
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0answers
33 views

Is there always $n$ permutations of a vector in $R^n$ that are linearly independent?

As long as the $n$ entries of the vector are all different and they dont add up to zero. If it is true, how to prove it, if not, what is a counter example?
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2answers
18 views

How many possible combinations/permutations?

I have 104 ingredients, and there are a maximum of 3 ingredients, how many recipes can be made? Take into account the order matters, that makes this a question of possible permutations instead of ...
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1answer
35 views

About the notation of composition of permutations in Lang's book

In Lang's "Algebra", p.30-31, I'm confused about the order of reading the composition of two permutations. In p.30, it seems that we read it from left to right (see the bottom equations), but for p.31,...
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1answer
27 views

How many solutions are there to an n by n queens problem?

Is there a way to calculate the number of solutions to n by n queen problem(done by backtracking) or it's complex and already defined as in the following table
2
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1answer
16 views

If $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$

Question In my group theory course, I am asked to show for $\sigma \in S_n$ that if $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$. My answer Let $\sigma \...
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1answer
20 views

Are all sets of $n$, s.t. $R(m)^n=I$, where $R(m)$ is any sequence of $m$ moves on a Rubik's cube and $I$ is the identity operator, known?

I've written a program that finds the number of times, $n$, one must apply any operation $R_i(m)$, which consists of $m$ single moves/turns/elementary operations on a Rubik's cube, s.t. $R_i(m)^n=I$, ...
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1answer
15 views

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite?

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite? Consider for instance a finite sequence of moves (...
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3answers
56 views

Every permutation is a product of two permutations of order 2

I am trying to solve a problem, not for homework, and it has me stomped! For $n\geq 4$ and $\alpha\in S_n$, $$\alpha=\dot{\alpha}\dot{\beta}$$ where $\dot{\alpha},\dot{\beta}$ are of order 2. I know ...
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1answer
25 views

Distribute 20 million $ among 4 companies with some constraints

20 million is to be invested in 4 companies A, B, C, D. The minimum amount for investments are 1, 2, 3, 4 million respectively. How many different investment strategies are available if An ...
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0answers
39 views

A kind of permutations and possible relation to cyclic groups.

Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ...
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1answer
39 views

In how many ways we can arrange 12 people in a row if 5 men are constrained to sit next to each other together?

In how many ways we can arrange $12$ people in a row if $5$ are men and they must sit next to each other? My approach I consider $5$ men as one entity and so now there are $8$ people to be seated ...
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3answers
32 views

Find the number of ways to arrange 8 students with restriction [on hold]

8 students are arranged in a row. How many ways to arrange them if 3 particular students must be separated?
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0answers
15 views

Discrete subgroups of $SU(m) \times SU(n) \times U(1)$

Is anything known about which permutation groups are subgroups of $SU(m) \times SU(n) \times U(1)$? Alternatively, how can one possibly go about finding them. I am trying to find discrete groups of ...
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5answers
747 views

How many possible “words” can be made from the first seven letters of the alphabet, allowing for repetition and enforcing alphabetical order?

Using letters from the alphabet $A = \{a, b, c, d, e, f, g\}$, how many words of length $5$ are possible when repetition is allowed but the letters must occur in alphabetical order? Not sure how to ...
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1answer
34 views

Permutations of geometric structure

Sorry about the title, i don't know how to describe this problem. I tried counting my way through this problem but kept getting the wrong answer(which is 12, by the way). Is there a more systematic ...
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0answers
31 views

Is there a method that determines an unknown permutation better than $\sum_{k=1}^n (k+1)/2$ steps on average?

Suppose I have a random permutation $s \in S_n$ that is unknown to me. However, suppose I can make a query where when I ask if $i$ is in the $j$th position in the permutation, I receive a yes or no ...
2
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1answer
92 views

Finding permutation with a condition

Let $a$ be a permutation in $S_6$. I'm asked whether there is an $a$ so that $a^2 = (123)(456)$ I'm quite confused about where to start. I do know the $a$ must consist of $3$ elements (right?). How ...
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0answers
38 views

what are the number of ways to select a 4 digit number with a 3 digit number always included? [on hold]

Number of ways to select 4 digit number( X X X X ) should have three digit number ( say 1 2 3 ) It should be in same order.
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1answer
70 views

Number of sequences that maintain a property

In how many ways can i create a sequence of $m$ elements from the set $1,2,...,n$ such that the longest strictly increasing subsequence of it is exactly $n$? For example if $n=3$ and $m=4$ then the ...
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2answers
24 views

Formation of Teams in Permutation and Combination

A class has $n$ students , we have to form a team of the students including at least two and also excluding at least two students. The number of ways of forming the team is My Approach : To include ...
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1answer
49 views

Elements that are their own inverses in a symmetric group.

How many elements are their own inverses in $S_6$? I'm having a hard time figuring out how to calculate such a thing.
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1answer
42 views

In how many ways can $5$ Indians, $4$ Chinese, and $3$ Americans be assigned to $12$ stations so that no two Americans serve at consecutive stations?

On a railway route from Delhi to Jaipur there are $12$ stations . A booking clerk is to be deputed for each of these stations out of $12$ candidates of whom $5$ are Indians , $4$ are Chinese and the ...
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1answer
36 views

counting number of steps using permutation-combination

We need to climb 10 stairs. At each support, we can walk one stair or you can jump two stairs. In what number alternative ways we'll climb ten stairs? How to solve this problem easily using less ...
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3answers
43 views

Identical Obects in Permutation and combination

There are $2$ identical white balls , $3$ identical red balls and $4$ green balls of different shades. The number of ways in which the balls can be arranged in a row so that at least one ball is ...
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1answer
25 views

Prove all products of two disjoint 2-cycles are pairwise conjugate in $A_{n}$

Let $n \geq 4$. Prove that in the group $A_{n}$ of parity-even permutations of $\{1,2,...,n\}$, all products of two disjoint 2-cycles are pairwise conjugate. This is a past exam question for a ...
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0answers
31 views

All 3-cycles are pairwise conjugate.

In my lectures on Group Theory our lecturer claimed the following: All 3-cycles are pairwise conjugate. He then went on to prove this but I am struggling with understanding his proof. I will try and ...
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0answers
14 views

Writing in disjoint cycles

Write $\left ( 123 \right )^{-1}\left ( 23 \right )\left ( 123 \right )$ in disjoint cycles. I get that $\left ( 123 \right )^{-1}$=$\left ( 132 \right )$ $\left ( 132 \right )\left ( 23 \right )\...
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2answers
31 views

Distinct digits in a combination of 6 digits

How many 6-digit numbers contain exactly 4 different digits? My approach is: For any 3 digis same and the remaining 3 different(aaabcd) 4*9*8*7*6 For any 2 duplicate digits(aabb) and the remaining ...
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2answers
66 views

How many 6-digit numbers contain exactly 4 different digits? [duplicate]

my solution is----> NUMBER can be 777210 this i denote by aaabcd ------ this can be done in ---> 10*1*1*9*8*7*[6!/3!] {1 for a thrice} NUMBER can be 772210 this i ...
2
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1answer
19 views

No subgroup of $S_n$ containing stabilizier of 1?

Is it true that the stabilizer of $1\in \left\{1,\dots ,n \right\}$ in $S_n$ is a maximal subgroup? Intuitively I'm thinking that as soon as you add another permutation, you'll somehow be able to ...
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6answers
35 views

Arrangement of 12 boys and 2 girls in a row.

12 boys and 2 girls in a row are to be seated in such a way that at least 3 boys are present between the 2 girls. My result: Total number of arrangements = 14! P1 = number of ways girls can sit ...
2
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0answers
30 views

Number of zigzag permutations of first $n$ natural numbers given start and end value

Given $n$ and $1\le s,e\le n$, how to compute the number of zigzag permutations of first $n$ positive integers starting with $s$ and ending with $e$? I tried formulating a recurrence relation but can'...
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0answers
95 views

How many ways can I arrange the numbers $1$ to $N$ with this divisibility condition?

For the numbers $1, \ldots, N$, how many ways can I arrange them such that either: The number at $i$ is evenly divisible by $i$, or $i$ is evenly divisible by the number at $i$. Example: for N = 2$...
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2answers
72 views
+250

Upper-bounding a sum over non-identity permutations

EDIT: Question 1 has been settled (below). The bounty is for question 2. Let $n\geq 3$ and consider the following function $f:S_n\backslash\{e\}\rightarrow \mathbb{R}$ $$f(\sigma)=\sum_{i=1}^n\frac{...
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2answers
28 views

Permutations & series [closed]

Consider all the $7$ -digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once, and not divisible by $5$. Arrange them in decreasing order. What is the $2015$th number (from the ...
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1answer
22 views

Concept of alike in Permutation and Combination

Number of ways in which $7$ green bottles and $8$ blue bottles can be arranged in a row if exactly $1$ pair of green bottles is side by side . (Assume all bottles to be alike except for the colour). ...
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1answer
28 views

Cyclic permutation group example ($n>1$)

I have googled around and haven't been able to find any examples of some $S_n$ with $n>1$ that is a cyclic group. This may mean it is a dumb question, any help is appreciated.
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1answer
32 views

How to find number of integral solutions, containing large number of cases?

Number of positive unequal integral solutions of the equation $x+y+z=12$ can be found out knowing the cases it involves: $(1, 2, 9) , (1,3,8), (1,4,7), (1,5,6), (2,3,7), (2,4,6) and (3,4,5)$. Thus, ...
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2answers
35 views

Permutation of 2 or more groups while keeping the ordering of the groups

I've been trying to get a general formula for this, but I couldn't find anything exactly what I need. What I want is, let's say we have 3 groups: (x,y,z),(a,b,c) and (k,l,m) What is the total number ...
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1answer
51 views

Understand a part of the proof about permutations in a symmetric group on $n$ elements

Let $\sigma$ be an even permutation in $S_n$($\sigma \in A_n$). Assume $\sigma = \tau\sigma\tau^{-1}$ for some $\tau \in S_n$ and assume that the type of $\sigma$ consists of distinct odd integers. ...
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22 views

Consider the system S which can take n input parameters and each parameter can take on m values

(a) What is the maximum number of pairs a single test case for this system can cover? "I know that there are m^n different combinations in this example, but i'm unsure how many pairs a single test ...
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1answer
45 views

Combination Problem : $6$ Countries , $4$ players from each country

$6$ Countries participate a world tournament . Each country has $4$ players. One Cricket player , One Rugby player , one Volleyball player and one Football player. Need to select a team of $8$ ...
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0answers
23 views

Summation containing permutations.

Given $$a_1< a_2<......< a_n$$ find a permutation $\sigma$ maximizing the sum $$\sum_{i=1}^n {a_i \over \sigma(i)}$$ I can't figure our where to begin. I know that the solution is $\sigma=...
0
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1answer
28 views

Permutation Product

Here is a problem from Algebra by Michael Artin: $p = \left( {\begin{array}{*{20}{c}} 3&4&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&5 \end{array}} \right),q = \left( {\...
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2answers
47 views

How many integers can be formed by using exactly $x$ $4$'s, $y$ $5$'s, and $z$ $6$'s if no other numbers are used?

Can anyone tell me the total number of integers than can be formed by using exactly $x$ $4$'s , $y$ $5$'s and $z$ $6$'s and no other numbers are used? For $x=1$, $y=1$, $z=1$, the total is $6 \...
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1answer
38 views

Can someone explain this proof of the relationship between chromatic number and independence number to me?

I came across the following claim and proof in this paper, and I really don't follow. If $G$ is a vertex-transitive graph with independence number $\alpha$ and chromatic number $\chi$ then $n/α(G) ≤...
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2answers
276 views

Counting: how many ways of climbing a stair?

You are climbing a staircase. At each step, you can either make $1$ step climb, or make $2$ steps climb. Say a staircase of height of $3$. You can climb in $3$ ways $(1-1-1,\ 1-2,\ 2-1)$. Say a ...
2
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1answer
22 views

Minimum number of moves (not swaps) to transform one permutation into another

I have an initial permutation (eg. $\{A,B,C,D,E,F,G,H~\}$) and a final permutation (eg. $\{A,C,F,D,E,G,B,H~\}$) and I want to find how the final permutation can be created from the initial one using ...
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2answers
31 views

permutation basic concept confusion

I have searched on many sites on internet but no one answered my question.. although question is not so different but as i believe in learning concept rather than memorizing it. I want to know why we ...