# Tagged Questions

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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### 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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### 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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### 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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### 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
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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 $\sigma \... 1answer 20 views ### Are all sets of$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$, ... 1answer 15 views ### Will$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}$is the identity operator, always be finite? Consider for instance a finite sequence of moves (... 3answers 56 views ### Every permutation is a product of two permutations of order 2 I am trying to solve a problem, not for homework, and it has me stomped! For$n\geq 4$and$\alpha\in S_n$, $$\alpha=\dot{\alpha}\dot{\beta}$$ where$\dot{\alpha},\dot{\beta}$are of order 2. I know ... 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...