Questions tagged [permutations]
For questions related to permutations, which can be viewed as re-ordering a collection of objects.
8,472 questions
0
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18 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 ...
3
votes
2answers
41 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 ...
0
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0answers
17 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 $15$ ...
1
vote
0answers
24 views
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 $15$ ...
0
votes
1answer
28 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 ...
1
vote
1answer
40 views
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
18 views
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,...
-1
votes
1answer
17 views
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?
-1
votes
2answers
40 views
Dihedral and permutation groups isomorphism [on hold]
Why dihedral group $\mathbb{D_4}$ is isomorphic with $\mathbb{S_4}$ and not isomorphic with $\mathbb{S_8}$?
-1
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2answers
35 views
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 $15$ ...
0
votes
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 ...
2
votes
1answer
48 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 ...
0
votes
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 ...
1
vote
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:
$\frac{17n^2-5n}{2}$
$...
0
votes
1answer
13 views
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 ...
6
votes
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 ...
2
votes
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
...
1
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0answers
58 views
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 ...
2
votes
0answers
18 views
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 $b$ ...
0
votes
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 ...
0
votes
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 ...
0
votes
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.
0
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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 ...
2
votes
2answers
48 views
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?
...
0
votes
1answer
36 views
Permutation in mathematica with one guaranteed element [on hold]
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 ...
0
votes
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 ?
0
votes
1answer
17 views
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 ...
0
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0answers
23 views
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.
0
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0answers
15 views
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_{...
-1
votes
0answers
33 views
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 ...
0
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0answers
19 views
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.
6
votes
1answer
232 views
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 ...
3
votes
2answers
94 views
$G$ 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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0answers
41 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, ...
4
votes
0answers
132 views
+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 ...
0
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0answers
20 views
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=\...
5
votes
1answer
88 views
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 \...
0
votes
1answer
46 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
0
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2answers
43 views
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 ...
1
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0answers
36 views
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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0answers
24 views
Solve the equation P(2n,3) = P(6,n) justifying that you have found all solutions [closed]
I am having some problems with solving this short exercise:
Solve the equation $P(2n,3) = P(6,n)$ justifying that you have found all solutions.
Thank you for your help.
Edit: e.g., by $𝑃(6,𝑛)$ I ...
2
votes
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 ...
0
votes
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 ...
1
vote
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.
1
vote
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....
0
votes
0answers
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 ...
1
vote
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.
...
0
votes
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:
<...
0
votes
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; ...
0
votes
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 ...
