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

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Output all permutations using 0-9 of of n-size up to 25. [Python]

Solving a problem in which I need to generate all possible permutations using the elements [0-9] of a number of n-size up o 25; where repetition is allowed. I've been using Python with different ...
0
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1answer
24 views

Find number of possible combinations from list of items [closed]

I have the following list: Product Icecream Banana Strawberry Vanilla The possible combinations (based on the spec) are 5: ...
0
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1answer
14 views

When to close brackets in product of disjoint cycle

when expressing 2 composition of function as a product of disjoint cycles, when do we 'close' the bracket? None of the sources explain this clearly. Some do not even make an attempt to.
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0answers
17 views

90 degrees CCW permutation of a square ($D_4$)

Given a square and a permutation of 90 degrees counter-clockwise, what is the order of this finite symmetry group? Here's an attempt: $$\rho =\begin{bmatrix} 1 &2 &3 &4 \\ \rho (1) ...
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1answer
27 views

large permutation question

What are the permutations of the following: 7 marbles each of 4 colors, for a total of 28 marbles. A 5x5 board, so 25 places for 1 marble to be placed. What are the permutations of placing the 25 ...
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1answer
38 views

A problem relating to combinatorics/ Permutation|Combination [closed]

The question is: (I actually may have messed up!) Suppose we have $4$ numbers namely : $a,b,c$ and $d$ Now we need to compare two of these numbers four at a time. For example : $a>b$ ...
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0answers
25 views

Counting number of ways in which switches can be pressed IF order doesn't matter

If there are given "K" number of distinct switches and "N" is any large number representing the total number of times those "K" switches will be pressed. We can easily say that the total number of ...
3
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1answer
17 views

The number of adjacent transpositions

If $\alpha\in S_{k+l}$, $\alpha=\left(\begin{array}{cccccc}1&\cdots&k&k+1&\cdots&k+l\\l+1&\cdots&l+k&1&\cdots&l\end{array}\right)$, for $k,l\in\mathbb{Z}^+$ ...
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1answer
23 views

Compute Square of Permutation

Dr. Pinter's "A Book of Abstract Algebra" presents the preface to a few exercises: In $S_{5}$, express each of the following as the square of a cycle (that is, express $\alpha^{2}$ where $\alpha$ is ...
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1answer
129 views

Oberwolfach Problem - 30 people at dinner on 3 tables of 10 seats each

There are $30$ people at an alumni dinner, seated at $3$ round tables of $10$ seats each. After every time interval $\Delta t$, a position change event is required where everyone changes position ...
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2answers
41 views

Representing a 5-cycle as a product of transpositions

Dr. Pinter's "A Book of Abstract Algebra" shows that: $$(12345)$$ can be written as the following product of transpositions: $$(54)(53)(52)(51)$$ How can the first representation, $(12345)$, be ...
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19 views

On permutation notation

I am trying to write up a proof that the parity of permutations on finitely many letters is well-defined. I think I have a proof that involves the number of disjoint cycles that a permutation may be ...
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0answers
35 views

Identity element of a group as a factorization of group elements.

For any group $G$ we readily verify that if $a,b,c\in G$ and $a*b*c=e$ , where $e$ denotes the identity element,then also: $b*c*a=e$ Indeed,let $b*c=x$.Then our problem amounts to: $a*x=e⇒x*a=e$ This ...
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2answers
73 views

Split a number into parts

In how many ways can a natural number $n$ be split into $m$ natural numbers (parts) where each part is less than $n$, the parts don't necessarily have to be equal, and all of them add up to $n$?
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2answers
70 views

seating arrangements of four men and three women around a circular table

In how many ways can $4$ men and $3$ women be arranged at a round tale if i)the women always sit together? ii)the women never sit together? I attempted both the questions but the answers I got don't ...
2
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1answer
39 views

A combinatorial question about outer automorphisms of $S_6$

Quite possibly I'll solve this and post my answer below, but maybe others will post better answers before I get to that. Or after.$^\dagger$ The group of permuations of $\{a,b,c,d,e,f\}$ is ...
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0answers
32 views

Conjugation of permutations

In the group $S_n$ I usually use the fact that if $(a_1 a_2 \dots a_r) \in S_n$ is an r-cycle and $\sigma \in S_n$ then $\sigma (a_1 a_2 \dots a_r)\sigma^{-1} = (\sigma(a_1)\sigma(a_2) \dots ...
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32 views

estimations in the birthday paradox?

The birthday paradox is the famous following problem: What is the probability $p_n$ that at least $2$ persons amongst $n$ persons chosen at random have the same birthday? Leap years are not taken ...
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1answer
42 views

Transitive action of the group implies isomorphism with the quotient by stabilizer

Let $\Omega$ be a set and $G$ a subgroup of the group $Sym(\Omega)$ of permutations of $\Omega$. Let $\omega \in \Omega$ and let $G_{\omega}$ denote the stabilizer of $\omega$ in $G$. If $G$ acts ...
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1answer
28 views

Combinations without Repetition [closed]

For calculating follow Combinations RRDD RDRD RDDR DRRD DRDR DDRR answer is $\frac{4!}{(2!\times2!)}$. First divide into $2!$ is because does not matter order ...
3
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1answer
53 views

Permutations of cards with no adjacent pairs

We have a standard 52-card deck, and are looking at the possible shuffles/permutations of this deck. However, we have rubbed off the suits from the cards, so for every rank (aces, tens, etc.) all ...
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1answer
72 views

What is the use and motivation for this particular concept in permutations?

Say you have the permutation $(54231)$ element of $S_5$ Now you drop say the "4" and then re-rank the remnant permutation on the other elements. Then you are left with, $(4231)$ element of $S_4$ ...
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0answers
18 views

Mixing time of three particle systems

Is there anything known about mixing time of Markov chains for three particle systems? It is proved here that the mixing time of an exclusion process is $\operatorname{O}(n)$. We can think if a ...
4
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1answer
32 views

Anagram with condition on last letter

How many ways can "computer" be arranged with a vowel as last alphabet? Isn't it $7! \times 3 $? since there are 3 vowels. $3$ (e,o,u) $ \times 7!$(number of arrangement without one of vowel). ...
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2answers
25 views

Permutation and Combinations Problem.

There are m copies of each n books on different subjects in the college library. The number of ways in which one or more books can be selected is ...?? I have no idea to deal with this problem , ...
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1answer
38 views

Natural action of $S_n$ on $\{ 1,2,\dots,n \}$

From reading online the "natural" action of $S_n$ on $\{ 1,2,\dots,n \}$ is $(g,x) \mapsto gx$. How is this action transitive? As far as I can see if we take $g$ to fix some element we will not get a ...
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2answers
83 views

How many 2 digit even numbers can be formed from these numbers?

How many even 2 digit numbers can be formed from the numbers 3,4,5,6,7? The digits cannot repeat (you can't have 44 or 66 for example). I know the answer to this is 8, because I just wrote them all ...
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2answers
28 views

More $1$s than $0$s in recursively defined set?

Let $S$ be the set of strings defined recursively by: Basis Step: $1 \in S$ Recursive Step: If $s \in S$, then $01s \in S$, $10s \in S$, $0s1 \in S$, $1s0 \in S$, $s10 \in S$, $s01 \in S$, $s1 \in ...
3
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1answer
21 views

How many permutations will there be to this problem?

How many permutations of the following pattern will there be. The order has to stay the same. In other words, you can only swap the 'B' with another 'B' because it will not affect the pattern? B C B ...
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1answer
58 views

Rolling 1 die 5 times [closed]

One die is rolled five times. How many different results are possible? Of those, in how many ways can there be exactly 2 rolls of 4? For the first part I multiplied 6 five times and got 7776. I'm ...
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2answers
31 views

Permutations with repetition element condition

I'm trying to figure out: How many permutations (with repetition allowed) does A,B,C have for a given $k$ (the length of the permutation) if A cannot be followed by a C anywhere in the end result? ...
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1answer
26 views

How to make 4608 combinations with these choices? (Probability, permutations/combinations)

This problem has been giving me a lot of trouble... Freeze King claims to offer 4,608 different ice cream cups. A customer can choose from 3 sizes, 4 flavors; a waffle cone, sugar cone, or cup; ...
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1answer
22 views

Prove this result relating to the sign of a permutation

Suppose that $\phi \in S_n$ is a permutation. Suppose also that $\psi = \phi \circ (i,j),$ where $1 \leq i, j \leq n.$ Why does it follow that sign$(\phi) = $ $-$sign$(\psi)$?
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1answer
26 views

How do I get number of combination for pairs of football teams?

Suppose we have 8 football teams playing each other in 4 matches. How do I find the number of combinations that is possible? E.g. Teams A,B,C,D,E,F,G,H can be in the following matches: Match 1: A vs ...
3
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1answer
25 views

Product of any two non disjoint cycles

Suppose you have a permutation $\sigma = \sigma_1 \sigma_2$ where $\sigma_1 = (i_1 i_2...i_k)$, $\sigma_2 = (j_1 j_2...j_l)$ and $i_{k_1},i_{k_2},...,i_{k_r}$ are equal to ...
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1answer
19 views

Is it possible to develop function that returns the number (rank, position) of a particular permutation.

I'm working on a data warehousing project and need to assign a unique value to a permutation and store that value as dimension in the data warehouse. Currently, I'm relying upon a rather large lookup ...
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1answer
39 views

Dividing $n$ identical things into $r$ groups

I was reading a course on Combinatorics where I came across following: The number of ways in which $n$ identical things can be divided into $r$ groups so that no group contains less than $m$ items ...
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3answers
56 views

Different ways of giving away 35 coins to 5 people?

The first part of the problem asks how many ways there are to give away 35 identical coins to 5 people, and I've concluded that it's ${35 \choose 5}$ because you're selecting how many ways you can ...
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1answer
42 views

How many different ways can you choose a group of 4 people?

You have a total of 9 people to choose from. Of these 9 people you are supposed to create a group of 4. How many different ways can the new group look? This is my reasoning: To the new group, the ...
3
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1answer
31 views

Generating Constrained Random Distributions

I am trying to help another StackExchange user. We are attempting to fill a 6x6 matrix with 12 A's, 12 B's, and 12 C's subject to the constraint that each row contains 2 A, 2B and 2 C and each column ...
3
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1answer
44 views

How to find a permutation of a specific rank?

I have a problem regarding permutations. When the rank of an unknown $S_7$ permutation is given, I want to find this permutation, but I can not. For example, I have the following questions: ...
0
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1answer
36 views

find the number of one-to-one function $[\pm n] \rightarrow [\pm n]$

the permutaion of $[\pm n]$ is a bijective (one-to-one) function $\pi:[\pm n] \rightarrow [\pm n]$ so that $\pi (-i) = -\pi(i)$ . $[\pm n]:=\{1, \dots, n-1, \dots, -n\}$. i have to find and determine ...
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2answers
47 views

Find a Four-element Abelian Subgroup of $S_5$ [duplicate]

Prof. Charles Pinter's "A Book of Abstract Algebra" provides this exercise: Ch 7 (Groups of Permutations) Part B #3 - Find a four-element abelian sub-group of $S_5$. Write its table. Please ...
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1answer
58 views

Which other “exotic” permutation-related things exist?

Some time back I posted some questions about the "exotic" outer automorphisms of $S_6$, and part of the answer was a citation of a paper by T. Y. Lam that said, among other things, that the ...
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4answers
99 views

How to write the set of all permutations on a set $n=\{1, 2, \ldots, n\}$

Let $n ∈ N$. Let $S_n$ denote the set of permutations on $\{1, . . . , n\}$. For any $σ ∈ S_n$, define $sign(σ) := (−1)^N$ , where $σ$ can be written as the product of $N$ transpositions. Now, let ...
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0answers
92 views

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$?

How many ways are there to fill up a $2n \times 2n$ matrix with $1, -1$ so that each column and each row has exactly $n $ $1$'s and $n$ $-1$'s ? I tried for cases $n=1 , 2$ but the solutions were ...
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1answer
43 views

Number of ways to place 4 girls into 3 bedrooms.

A family has 4 girls and 3 bedrooms. 2 of the bedrooms are only big enough 1 girl, and the last room is big enough for 2 girls. How many ways are there to assign the girls to the bedrooms? I came up ...
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2answers
82 views

Exotic maps $S_5\to S_6$

This section says: There is a subgroup (indeed, $6$ conjugate subgroups) of $S_6$ which are abstractly isomorphic to $S_5$, At this point I'm thinking: certainly: the group of all ...
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0answers
60 views

How many possible six-word sentences

A word is defined as a nonempty (possibly meaningless) sequence of letters. How many $6$-word sentences can be made using each of the $26$ letters of the alphabet exactly once? Generalise the result ...
4
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1answer
71 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...