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Questions tagged [permutations]

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

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Denoting sets (permutations,tuples?),

I'll use and example to explain my question: Example: Let there be two distinct types $a$ and $b$ and each type has equal number of sets. There are two sets of type $a$: $\{1,2\}$ and $\{3,4\}$. ...
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26 views

Find a formula for number of orbits under action of $D_{4}$

We colour each side of a square with $k \geq 1$ colours. Find a formula for the number of orbits under the action of $D_{4}=\{ e , r,r^{2},r^{3},s,sr,sr^{2},sr^{3} \}$ on the set of colours. Now as ...
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2answers
50 views

Probability of nth draw being same value

Disclaimer at the beginning: this does relate to a homework problem, but is not the actual homework problem itself. Consider a series of numbers, $1, 2,...,n$. We select one value at a time at ...
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21 views

Permutations of alike objects 3

If I have $1$ jar of $20$ balls with $10$ red, $5$ blue and $5$ yellow, $1$ jar of $30$ balls with $10$ red, $10$ blue and $10$ yellow and $1$ jar of $50$ balls with $15$ red, $20$ blue and $15$ ...
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Permutations of alike objects 2

If I have $1$ jar of $20$ balls with $10$ red, $5$ blue and $5$ yellow, $1$ jar of $30$ balls with $10$ red, $10$ blue and $10$ yellow and $1$ jar of $50$ balls with $15$ red, $20$ blue and $15$ ...
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1answer
30 views

Independence of Records of Permutation

For any permutation $\sigma = (\sigma_1,\dots,\sigma_n)$, call $\sigma_k$ a record if $\sigma_k>\sigma_i$ for $1\le i\le k-1$. Let $\alpha_k$ be the indicator for the event that $\sigma_k$ is a ...
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1answer
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How to understand the structure of the interesting graph obtained from the group?

Let $G = A_5$ and $H < G$ is subgroup of $A_5$ generated by $(12)(34)$, and $(125)$. Define graph $\Gamma$ by a vertex set is element of $G$ and elements $x$ and $y$ adjacent if $|H^x \cap H^y| =...
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4answers
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Number of permutations with cycle shape (2,2) in S5

I probably need permutations of form $(ab)(cd)$. If I had a 3 cycle $(abc)$ I would use something we call arrangements. For example arrangements of $n$ choose $k$ is just $\frac{n!}{k!}$. In this case,...
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1answer
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Formula for permutations and combinations [on hold]

Suppose there are $27$ possible letters $\{A, B,... ,Z\}$ and they can be arranged in any order including spaces, capitalization, order. Is the formula $P(n,r) = n!(n−r)!$ still correct?
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2answers
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Dihedral and permutation groups isomorphism [on hold]

Why dihedral group $\mathbb{D_4}$ is isomorphic with $\mathbb{S_4}$ and not isomorphic with $\mathbb{S_8}$?
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Permutations of alike objects [duplicate]

If I have $1$ jar of $20$ balls with $10$ red, $5$ blue and $5$ yellow, $1$ jar of $30$ balls with $10$ red, $10$ blue and $10$ yellow and $1$ jar of $50$ balls with $15$ red, $20$ blue and $15$ ...
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2answers
29 views

Permutation from left to right of $140$ objects

I've been stuck on this problem for quite some time now, I can't seem to find a video or such where it references permutations in a specific order of left to right. I have no idea how to set this ...
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1answer
49 views

Finding a set on which a group acts on

I have a given group, defined by its table, namely: Now I am asked to find a set $X$ where this group acts upon non trivially. I have problems understanding this question. As I understand, I am ...
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1answer
21 views

permutations cycle

I am doing abstract algebra problems, but unfortunately, the book we are using for the class is quite poor and leaves out lots of definitions and explanations, so I am not even sure if I completely ...
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2answers
41 views

Number of ways of selecting two integers $a$ and $b$ from the set $\{1,2, 3, … ,5n\}$, $n∈ N$ so that $a^4 – b^4$ is divisible by $5$ [on hold]

What is the number of ways of selecting two integers $a$ and $b$ from the set $\{1,2,3,\ldots ,5n\}$, $n \in N$ so that $a^4 – b^4$ is divisible by $5$? Here are the options: $\frac{17n^2-5n}{2}$ $...
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1answer
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How many different arrangements are possible with 3 digits numbers from 1-9 if we have to arrange them in ascending order?

if there are numbers 1,2,3,4,5,6,7,8,9 how many different three digit values we can make such that the digits are in ascending order? since there are numbers from 1-9, I understand that the first ...
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2answers
41 views

What would be the number of inequivalent $6$-colourings of the faces of a cube?

Consider the different ways to colour a cube with $6$ given colours such that each face will be given a single colour and all the six colours will be used. Define two such colourings to be equivalent ...
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2answers
42 views

What is the difference between these two combinatorics problems?

So the first problem is "In how many ways can we arrange the letters in the word Alabama." and the second questions is "In how many ways can we arrange three Mathematics books, five English ...
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0answers
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Permutation and Combination/ Counting

Q. Our student clubs are being particularly active. The aquatics club in particular is in training for an upcoming competition. For all parts of this question, working is required, including ...
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Permutation of multiple object types with one type kept in a fixed range

We have to place 50 objects of 3 types in a row with 50 places. There are 26 objects of type $a$, 16 objects of type $b$ and 8 objects of type $c$. I am working on a problem where objects of type $b$ ...
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1answer
26 views

Number of sortable permutations of a set of natural numbers. [on hold]

Define the permutation of the set $\text{{1,2,3,...n}}$ to be sortable if upon cancelling appropriate term of such permutation remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of ...
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1answer
17 views

Counting members of equivalence classes of injective functions

Let $A=\{f:\{1,2,3,4\} \to \{1,2,3,4,5,6,7,8\}, f$ injective$\}$. Let $R$ be a relation on $A$ such that $f_0Rf_1 \iff f_0(1)+f_0(2) = f_1(1)+f_1(2)$. If $h \in A, h(n)=n+2$, what's the cardinality of ...
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1answer
11 views

Number of times we have to compose a permutation in order to have exactly k fixed points

Let $f = (1 4 6)(2 7 5 8 10)(3 9)$ in $S_{10}$. Find an integer $n$ such that $f^n$ has exactly $7$ fixed points. I provided the exact numbers, but would welcome a more general solution.
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1answer
36 views

Demonstration with Vandermonde's Identity

VANDERMOND’S IDENTITY: Let m, n, and r be nonnegative integers with r not exceeding either m or n. Then $${m+n \choose r} = \sum_{k=0}^{r}{ {{m}\choose{r-k}} { {n} \choose {k} } }$$ The exercise ...
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2answers
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Why are these two ways of counting the same combinatorial object not yielding the same result?

In how many ways can we distribute 20 black balls and 2 white balls (indistinguishable modulo color) in 5 numbered glasses such that the fifth one doesn't have more black balls than white balls? ...
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1answer
36 views

Permutation in mathematica with one guaranteed element [closed]

I just want to create a Permutation-List of elements {a,b,c,d,...} where each Permutation of three elements needs to contain at least one specific element ...
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0answers
20 views

Ways in which circles cut each other [closed]

What will be the number of ways n circles cut each other, for n like 2,3 it is doable but is there any pattern which can be used to build a program or some sort of a series to follow up ?
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1answer
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Permutations: Notation that uses composition with exponents?

Notation question. Wiki Compositions of permutations claims that compositions : $$\sigma \bullet \pi$$ is the function that maps any element $x$ of the set to: $$\sigma (\pi (x))$$ and that ...
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In how many ways can n men and n women sit in n seats serially, so that no 3 women sit consecutively [duplicate]

I tried a lot, but couldn't find a way. One thing which I found interesting is that for n = 7, the maximum women can be W W M W W M W = 5. Hope this might help someone to give an answer.
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Discret optimization problem

the problem setting is as follows: We are given a set of words (for simplicity we can assume all of the same length, i.e. n-tuples) $W = \{ w_1, w_2, ..., w_N \}$ where $w_i = (a_{i1}, a_{i2}, ... a_{...
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Find the number of subsets such that sum of the elements in each subset is divisible by $9$ [duplicate]

Given $A=\left\{1,2,3,\cdots 2019\right\}$, Find the number of subsets of $A$ such that sum of the elements in each subset is divisible by $9$ My try: I was trying a very long way but a basic ...
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How many arrangements of the letters of the word REMEMBRANCE in a line do not have all four vowels next to each other? [closed]

The 11 letters of the word REMEMBRANCE are arranged in a line. Find the number of different arrangements which do not have all 4 vowels (E,E,A,E) next to each other.
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1answer
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How does a permutation act on a string?

Is there a conventional way to have a permutation act on a list of objects? It seems like there are two possible ways, one being the inverse of the other. Suppose I have a permutation $\sigma \in ...
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2answers
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$G$ is an abelian group, and a permutation $T$. Suppose that $x-T(x)\neq y-T(y)$ for all $x\neq y$. Show that $p(x)=x-T(x)$ is also a permutation

Let $G=(A,+)$ be an abelian group, T a permutation on $A$. Assume that $x - T(x) \neq y - T(y)$ for all distinct $x,y \in A$. Show that $p(x)=x - T(x)$ is a permutation on $A$. For that I would show ...
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43 views

Prove: Permutation of a root is another root of a polynomial

I read that Galois group is a permutation of the zeros or roots, this is new to me, so, I have a question. How can I prove, all roots of a polynomial are permutation of one another? in other words, ...
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+50

Bipartite graphs from permutations

Given are $n\geq 1$ permutations of $abcd$. We construct a bipartite graph $G_{a,b}$ as follows: The $n$ vertices on one side are labeled with the sets containing $a$ and the letters after it in each ...
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Confusing the order of two product of permutations [duplicate]

Let $\sigma$ and $\tau$ be distinct(means relative prime) permutations of order $m$ and $n$, respectively(need not be $\gcd(m,n)=1$) in certain symmetric group. Now, suppose that $\sigma\circ\tau=\...
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1answer
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Find a permutation of the rows of a matrix that minimizes the sum of squared errors

I'm struggling with the following problem: Let $A, B \in \mathbb R^{n \times d}$. Denote by $\mathcal{P}$ the set of all possible permutations of the rows of $A$. Find a permutation $\pi \in \...
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1answer
47 views

How many are the anagrams of the word MISSISSIPPI in which there are no two consecutive I letters? [closed]

Can anyone give me a complementary approach to this problem? That is, the calculation of the total minus the "unwanted". Answer: 1050
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2answers
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How many strings of six lowercase letters from the English alphabet contain the letters *a* and *b* with all letters different?

How many strings of six lowercase letters from the English alphabet contain the letters a and b with all letters different? Answer: 7650720 I saw a resolution that considered the two types of strings ...
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Counting the permutations of a set in which some elements can be repeated

Suppose I have $n$ cards that I've arbitrarily placed into $p$ piles, possibly of different sizes. The cards are available in $c$ colors, and all cards with the same color are considered identical. ...
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1answer
250 views

How many 3-word sentences can be created?

I'm doing this old exam in my course and I stumble upon the question: How many $3$-word sentences can be created from $8$ letters A, $8$ letters B and one of each of the letters C,D,E,F,G when each ...
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1answer
26 views

In how many ways can $12$ children occupy the six banks of two seats on a Ferris wheel?

1) In how many ways can $12$ children occupy the six banks of two seats on a Ferris wheel? (consider important the order of two children sitting in a given bank). Resp.: 7680 2) What would be the ...
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2answers
101 views

How many are the non-negative integer solutions of $𝑥 + 𝑦 + 𝑤 + 𝑧 = 16$ where $x < y$? [closed]

How many are the non-negative integer solutions of $𝑥 + 𝑦 + 𝑤 + 𝑧 = 16$ where $x < y$? Anyone can explain how to think to approach this type of problem? The answer is 444.
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2answers
30 views

How many six-letter words can be formed with the letters of the word ‘policy’ such that the vowels can only occur in even positions?

We have $3$ places for evens and two vowels so selecting $2$ vowels for $3$ positions would be $3P2 =6$ but how can I do the same thing for consonants. Now I've left with $3$ places and $4$ consonants....
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39 views

Galois Theory: Quartic and Klein 4 group

I'm studying Galois Theory on my own and already understand the correspondence between field automorphisms and subgroups of the permutation group. However, I am trying to approach it more from the ...
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1answer
28 views

How do I generate permutations consisting of all elements in multiple ordered sets (preferably incorporating element priority)?

I am doing some self study and have come up with a problem I'm not sure how to go about solving. Please correct me if I've used any term incorrectly. Purely mathematical or Python approaches are fine. ...
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1answer
29 views

Number of partitions of list [closed]

If I have my_list = [0, 1, 2]. I want to figure out how many possible list partitions there are for a list of length $n$. For the above example, there would be: <...
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0answers
50 views

balls and boxes; permutation and combination.

There are $2010$ boxes labeled $B_1, B_2, . . . . B_{2010}$ and $2010n $ balls have been distributed among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves; ...
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2answers
85 views

Count the number of permutations of certain cycles type

Suppose we have $n$ elements, assume there is a permutations over $k$ elements among the $n$ elements so $n-k$ are fixed. Let that the permutation over the k elements is represented by permutation ...