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

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how many different ways a set of ten variables could be ranked from highest to lowest value, given three possible integer values for each variable

I'm trying to solve a problem involving how many different ways a set of ten variables could be ranked from highest to lowest value, given three possible integer values for each variable. It's been ...
7
votes
4answers
693 views

Prove this using counting techniques: $\sum_{k=0}^{n}{\binom{2n+1}k} = 2^{2n}$

I recently came across a question while studying for an exam. I haven't been able to solve it. We had to prove: $$\sum_{k=0}^{n}{2n+1\choose k} = 2^{2n}$$ We had to use counting techniques. This was ...
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0answers
20 views

Formulate as standard problem: Optimal cyclic permutations

How can we find cyclic permutations $\prod_i$ to be applied to each of corresponding $i$'th rows of a square matrix $X$ of size $n \times n$ such that a given sum of pairwise costs $\sum_{ij}C\left[\...
2
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0answers
21 views
+50

How to find the average Kendall's distance between 2 rankings

Suppose I have 2 rankings: $1$, $2$, $3$ and $2, 1, 3$ then the Kendall's distance between the two is 1 since there is only one pairwise adjacent switch. My question is, suppose my 2 rankings each ...
2
votes
0answers
88 views

Conjugacy class $A_4$

I want to find all conjugacy classes of $A_4$. So basically what I did, I took all elements of $A_4$ and calculated their conjugates. I had no problems with $$\{e\}, \{(123),(134),(142),(243)\}, \{(...
3
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1answer
66 views

Team grouping troubles

Imagine there are 12 teams, numbered 1 through 12. There are 10 games those teams can compete in, with two teams needed per game. There are 10 rounds, and it is important that after the 10 rounds are ...
1
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1answer
402 views

Circular Nonconsecutive Permutations

A carousel has eight seats, each representing a different animal. Eight girls are seated on the carousel facing forward (each girl looks at another girl's back). In how many ways can the girls change ...
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0answers
16 views

Solve $\max \mathrm{sum}(AXB \geq \gamma), X \in \{0,1\}^{N \times N}$

I have a problem to find the best permutation matrix $X \in \{0,1\}^{N \times N}$, so as to maximize the number of elements in $AXB$ which are above a certain positive number $\gamma$. In other ...
1
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2answers
64 views

Find the divisors of $5040$ in the Plato's dialogue “Theaetetus”

In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem" \begin{align*} 5040 &= 7! \\ &= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\ &= 2 \...
0
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1answer
28 views

Sylow subgroup of a symmetric group

Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which ...
0
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0answers
25 views

Number of distinct integer-valued vector solution for $x_1 + x_2 + … + x_r = n$ [duplicate]

The Number Of Integer Solutions Of Equations $$x_1 + x_2 + ... + x_r = n$$ An approach is to find the number of distinct non-negative integer-valued vectors $(x_1,x_2,...,x_r)$ such that $$x_1 + x_2 +...
3
votes
3answers
363 views

Arranging numbers around a square

In how many ways numbers 1 to 12 can be arranged on a sides of squares (5 places on each sides i.e 20 places total) leaving 8 places empty? I am getting answer as 12c5(selecting 5 numbers)*7c5(...
0
votes
0answers
26 views

Shuffles vs direct sums of permutations

A $(p,q)$-shuffle is a permutation of $p+q$ things that preserves the internal order of the first $p$ things and of the last $q$ things. As remarked on wikipedia, since a $(p,q)$-shuffle is uniquely ...
2
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1answer
51 views

Black are berries and maroon are cherries. Place 8 more cherries removing berries 1 from each row and each column. No of ways?

I tried to see it as a matrix where for a position (i,j) , i+j = 8, 9, 16 means you can't change that position. Any help?
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1answer
26 views

Question of P &C [on hold]

There are 3 pots and 3 coins. All thesecoins are to be distributed into these pots where any pot can contain any number of coins. In how many ways all these coins can be distributed such that no pot ...
0
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1answer
26 views

Tournament Of The Towns King and the 1000 wizard's

So i was doing one of the question's of TOURNAMENT OF THE TOWNS and I was not able to understand the solution given by them. The problem is: The King decided to reduce his Council consisting of ...
3
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2answers
29 views

How many elements in $S_{8}$ are conjugate with $(12)(345)$?

How many elements in $S_{8}$ are conjugate with $(12)(345)$? My reasoning is as follows: Two elements in $S_n$ are conjugate if and only if they have the same cycle type, so we need to count the ...
-2
votes
4answers
94 views

Permutation in group theory [on hold]

I am confuse how to proceed for the following question. Can you please help me. Thanks in advance! For a permutation $\pi$ of $\{1,\cdots,n\}$, one say that $k$ is a fixed point of $\pi$ if and only ...
2
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3answers
22 views

Probability of increasing order permutation

Suppose I have n elements. What's the probability of a permutation such that the first half is increasing and second half can be ordered without any constraints? (A permutation can only have distinct ...
2
votes
2answers
50 views

How to find the N-th 3 word sequence within the following constraints

I have a list of words. Let's say that I have an algorithm(explained below) to generate the permutations in a specific order. I want to be able to find the N-th permutation easily. I want to make ...
9
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2answers
799 views

Why is some power of a permutation matrix always the identity?

If you take powers of a permutation, why is some $$ P^k = I $$ Find a 5 by 5 permutation $$ P $$ so that the smallest power to equal I is $$ P^6 = I $$ (This is a challenge question, Combine a 2 ...
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1answer
35 views
3
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3answers
99 views

Number of positive unequal integer solutions of $x+y+z+w=20$

What is the number of positive different integer solutions of $x+y+z+w=20$, where $x,y,z,w$ are all different and positive? It would be nice if coding is not used. I am given the answer $552$.
3
votes
3answers
47 views

How to express/write a permutation of a Set?

How to express a permutation (without repetition) of a Set $A$? I'd like to create a set $P$ of tuples while equal tuples should only occur once in the set $P$. Tuples are equal when e.g. $\{a, b\} = ...
6
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3answers
371 views

An epimorphism from $S_{4}$ to $S_{3}$ having the kernel isomorphic to Klein four-group

Exercise $7$, page 51 from Hungerford's book Algebra. Show that $N=\{(1),(12)(34), (13)(24),(14)(23)\}$ is a normal subgroup of $S_{4}$ contained in $A_{4}$ such that $S_{4}/N\cong S_{3}$ and $...
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2answers
32 views

Number of permittable numbers given following conditions.

What are total numbers belonging to $\mathbb Q$ (rational) between $2008$ and $2009$ such that after decimal point their digits occur in decreasing order? \begin{align} 1) &\ 9Pi;i\in [1,9], \\ 2)...
1
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5answers
64 views

Arrange black and white balls so that each pair of white balls is separated by at least two black balls

I am trying to solve the following question: How many linear arrangements of $m$ white balls and $(n-m)$ black balls are possible such that each pair of white balls is separated by at least two ...
1
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1answer
25 views

Betting ended after nth round.Find the sum of money NOT WON?

Rahul and Vijay are playing a game with 12-sided die,where both of them lay bets on outcomes of roll of die.They start betting Rs 5 each on first round of the game and the amount bet in each ...
39
votes
5answers
2k views

Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of ...
0
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1answer
52 views

How to approximate the Langford numbers with probability?

A Langford pairing, also called a Langford sequence is a permutation of the multi set {$1,1,2,2, \dots, n,n$} in such a way that there are exactly $k$ elements in between every $k$. Interestingly, ...
0
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1answer
41 views

Monotone subsequence in a random permutation

I wish to compute the probability of having a log(n) length consecutive monotone subsequence in a random permutation of {1,...,n} (log with base 2). I'm trying to show it's $\leq1/n$, does it make ...
0
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1answer
60 views

Probability for a random permutation

Given a set of $N$ elements and a uniformly distributed random number generator, which always generates values between $0$ and $N-1$. Then the probability to get a random permutation (without re-draws)...
1
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1answer
23 views

Permutations (Making numbers from digits) [closed]

Using the digits 1, 2, 4, 5, 7, and 8, how many different three-digit numbers can you form if each digit may be repeated any number of times in a number? I have tried to do this question and tried ...
2
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0answers
35 views

Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
0
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1answer
31 views

How would I calculate the total number of combinations [closed]

Lets say I have 4 lines or rows lets call them Row 1 .. Row 4 Now the total number of ways to delete the rows are: Row 1 (leaving Row2, Row3, Row4) Row 2 (leaving Row1, Row3, Row4) Row 3 Row 4 ...
3
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1answer
34 views

Generic method to distribute n distinct objects among r people such that each person gets at least one object

Is there any generic method to solve problems of the kind - "How many ways to distribute n distinct objects among r person(s) such that each person gets at least 1 object?". I am aware of 2 different ...
16
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2answers
616 views

Minimal generating set of Rubik's Cube group

The Rubik's Cube group is generated by the six moves $\{F,B,U,D,L,R\}$. However, is this the minimal generating set for the group? In other words, can I simulate the move $F$ just by making the moves $...
0
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1answer
34 views

Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
0
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1answer
20 views

N objects among K persons

In how many ways can we distribute N objects among K people such that each person recieves AT LEAST ONE object ? Also the SUM MUST BE EQUAL TO N. eg. 7 objects can be distributed among 5 people in 2 ...
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2answers
25 views

Permutation of coefficient with coditions [closed]

I have 6 coefficients, (V1,V2,H1,H2,D1,D2). Their permutation is 6! = 720. But I have a rule: V2 cannot lead V1, H2 cannot lead H1 and D2 cannot lead D1. For example: V2V1H1H2D1D2 is prohibit. ...
3
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1answer
39 views

Permutations of $S_7$

Find all permutations $\alpha \in S_7$ such that $\alpha^3 = (1 2 3 4)$. My attempt: We know that such an $\alpha$ must "look like" $(1432)$, since $(1432)^3=(1234)$. I think I need to find the ...
5
votes
1answer
29 views

Number of $4\times3$ matrices of rank 3 over a field with 3 elements.

I am finding number of $4\times3$ matrices of rank 3 over a field with 3 elements. If i count it as number of linearly independent columns i.e $3$ then its answer is $(3^{4}-1)(3^{4}-3)(3^{4}-3^{2}).$ ...
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1answer
63 views

Number of permutations with two elements in one cycle

Show, that for a set of permutations of a set $\{1,\dots,n\}$ $(n>0)$ the following statement is true. statement: The number of permutations where $1$ is in the same cycle with $k$, and the number ...
3
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1answer
28 views

Bell numbers and the Moments of expected number of fixed points

Let $X_N$ be the random variable corresponding to the number of fixed points (1-cycles) in a permutation chosen uniformly at random from $S_N$. Then, the $m^{\text{th}}$ moment, when $m < N$, is ...
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3answers
31 views

Finding all normal subgroups of $A_4$

I was reading up on this: Find the number of normal subgroups of $A_4$. If $H$ has a $3$-cycle, say $(123)$, then $H$ has its inverse $(132)$ thefore it also has $(124) = (324)(132)(324)^{-1}$, ...
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0answers
23 views
1
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1answer
26 views

20 identical balls to be distributed in 3 identical boxes with MAX & MIN balls in each box?

As the title suggests, In how many ways can 20 identical balls be distributed in 3 identical boxes with at most 8 balls in each box and minimum 1 ball in each box ?
1
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2answers
51 views

Scheduling a Round Robin tournament - 4-way games

I'm looking to schedule 16 players to play a round robin tournament with each other such that there are 4 players at each table. I'd like for each player to play with each other player exactly once ...
1
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1answer
46 views

Notation for probability: $C_n^r$, $P_n^r$, $A_n^r$?

I was told that $C^{n}_{k}$ refers to combinations or choose k elements from n elements, $\bar{C^{n}_{k}}$ refers to combinations with repetitions (i.e. $C^{n+k-1}_{k}$), and $P^{n}_{k}$ refers to ...