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

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All equivalent moves on a rubik's cube

Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively. Let us say that I have a permutation of the stickers on the cube written as a word ...
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1answer
40 views

Subgroups of $G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}$ [on hold]

Let $$G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}.$$ Is this a subgroup? By examining particular examples, one can see that it is not, since it is not closed under composition. However, ...
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0answers
21 views

Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group ...
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1answer
34 views

Order and generators of intersection of cyclic groups

Let $\sigma$, $\tau$ be two permutations of $S_n$. We know that $\langle \sigma \rangle \cap \langle \tau \rangle$ is a cyclic subgroup and (from Lagrange's Theorem) that its order divides ...
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18 views

What is the probaility that two random permutations have same order?

I am interested in the orders of random permutations. Since the law of the log of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), one ...
1
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1answer
31 views

The probability of being dealt at least 5 wanted cards

In a fictional deck of cards, there are 30 cards, 15 different ones (each card has an identical pair, so 15 pairs = 30 cards). I want to answer the question: I am dealt 10 cards. I wish to receive 5 ...
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1answer
24 views

Subgroups of $S_{10}$ that contain $\langle \sigma = (123)(456)\rangle$

In $S_{10}$ let us consider the permutation $\sigma = (123)(456)$ and the cyclic subgroup $ G = \langle \sigma \rangle$. I can fairly easily find that $$\langle \alpha = (1,4,2,5,3,6)(7,8) \rangle ...
3
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1answer
15 views

Multi-stage Probability

I think the easiest way to explain what I'm having trouble with is to give an example question: A monkey is given 12 blocks: 3 Squares, 3 Rectangles, 3 Triangles, 3 Circles. Calculate the probability ...
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1answer
14 views

find all combination without overlapping

(*Constraint) Kinds of number should be limited in 3 (ex. {1,1,1,1} o/ {1,2,3} o / {1,1,2,2,5,5,4} x) And I want to find series of Integer. For example, if n = 4 (n is length of numbers) {2, 7, 2, ...
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1answer
23 views

Probability of drawing $m$ of $A$ in $n$ cards given a deck of $d$ cards contain $a$ copies of $A$?

As in the title I'm trying to work out what the chances of drawing $m$ copies of a specific card in $n$ draws are given a deck size of $d$ containing $a$ copies of $A$. I've tried using permutations ...
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2answers
30 views

Find the number of seating arrangements at a round table of three single men, two single women, and two families

Three single men, two single women and two families take their places at a round table. Each of the two families consists of two parent and one child. Find the number of possible seating arrangements. ...
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1answer
18 views

Number of permutations of $S_n$ such that $\sigma^h(a) = \sigma^k(b)$

A basic result in combinatorics is: In $S_n$ there are $(n-d)(n-2)!$ permutations $\sigma$ such that $\sigma^k(a) = b$, if $a \neq b$; $d(n-1)!$ permutations $\sigma$ such that ...
2
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2answers
39 views

In how many ways can a committee of $6$ people be selected from $7$ men and $6$ women if it can contain at most one of persons A and B?

A committee of $6$ people will be formed with $7$ men and $6$ women. The oldest of the $7$ men is A and the oldest of the $6$ women is B. It is described that the committee can include at most one of ...
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0answers
31 views

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$

Prove that if $\tau \in N_{S_A}(H)$ then $\tau$ stabilizes the sets $F(H)$ and $A \backslash F(H)$ $H$ is the set of fixed points on $A$ $A$ : set, $H \le S_A$, $F(H) = \{ a \in A : \sigma (a) = a, ...
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1answer
21 views

Prove the set of permuations which permute only finitely many elements is a normal subgroup

Let $A$ be a non-empty set and let $X$ be a subset of $S_A$ Now let $F(X) = \{a \in A : \sigma(a) = a, \forall \sigma \in X\}$, $M(X) = A\backslash F(X)$, and $D = \{ \sigma \in S_A : \mid ...
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2answers
22 views

Probability; bridge hand question

$13$ cards are chosen at random with no replacement from a deck of $52$ cards. find the probability there are $5$ spades chosen, $4$ hearts, $3$ diamonds and $1$ club. I got ...
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1answer
34 views

Properties of lists with arbitrary lengths and alphabet size

I am having trouble understanding a problem of applying the concepts of permutations and combinations in an example that I found while reading my textbook. Basically, it wants the number of elements ...
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3answers
27 views

How many different varieties of pizza can be made if you have the following choices:

How many different varieties of pizza can be made if you have the following choice: small medium, large; thin, hand tossed, pan; and $12$ toppings (cheese is an automatic), from which you may select ...
3
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1answer
47 views

How can I check whether the group $[16,13]$ in GAP with $3$ generators can be generated by $2$ elements?

The group $[16,13]$ in GAP has structure $(C4\times C2):C2$ and is generated by the permutations $(1234)(5678)$ , $(15)(26)(37)(48)$ and $(57)(68)$ . The group $[16,3]$ in contrast with the same ...
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0answers
13 views

Fraction of permutations satisfying a poset

Let $[n]:=\{1, ..., n\}$. Let $P$ be a poset on $[n]$. What is the fraction of permutations that satisfy $P$ when we view a permutation as inducing a linear ordering on the numbers? For instance, if ...
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21 views

G-set of 8 colors and 6 sides

"How many distinguishable wooden cubes can be painted if u use 8 colors (different colors on every side)" I have solved this question using Burnside's lemma ...
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1answer
46 views

Prove that (01) and (01…n-1) generate Sn? [duplicate]

Show that every element of Sn can be written as an arbitrary product of the elements (01) and (01...n-1). I understand that this can be solved using induction, and I've set up my base cases. ...
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2answers
45 views

What is the number of ordered triplets $(x, y, z)$ such that the LCM of $x, y$ and $z$ is …

What is the number of ordered triplets $(x, y, z)$ such that the LCM of $x, y$ and $z$ is $2^33^3$ where $x, y,z\in \Bbb N$? What I tried : At least one of $x, y$ and $z$ should have factor ...
6
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3answers
153 views

Is there any algorithm to find Isomorphism function between two graphs?

Two graphs which contain the same number of graph vertices connected in the same way are said to be isomorphic. Formally, two graphs and with graph vertices are said to be isomorphic if there is a ...
3
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1answer
42 views

How many DFA's exist with two states over the input alphabet $\{0,1\}$?

How many DFA's exist with two states over the input alphabet $\{0,1\}$? My attempt : Input set is given. So, we have 3 parts of DFA which we can change: Start state Transition Function Final ...
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0answers
35 views

$S_X = \{f(x) = x : \text{ bijective} \}$. prove $S_x$ is isomorphic to $S_n$

Aright, to start $S_n$ is the Symmetric group and $S_X = \{x_1, x_2, \ldots x_n\}$. Going through the mapping $\phi(S_X) \to S_n$, I'm not sure how I'd show this mapping and the first thought that ...
5
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2answers
122 views

How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...
4
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1answer
58 views

Alternating group on infinite sets

It is well known that the only normal subgroup of $S_n$ is $A_n$ when $n\geqslant 5$, and that $A_n$ is also simple. Furthermore, $A_{\infty}$, the even permutations on $\mathbb{N}$, is also simple. ...
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0answers
22 views

oligomorphic subgroups of $S_\infty$

Is it true that every oligomorphic subgroup of $S_\infty$ is not abelian? A subgroup of $S_\infty$ is said oligomorphic if its action on $\mathbb N^n$ has only finitely many orbits for each ...
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1answer
36 views

Polish subgroups of $S_\infty$

Let $S_\infty$ considered as Polish Group. Prove that every Polish subgroup of $S_\infty$ has the following form: $\overline{{\left \langle X \right \rangle}}$, where $X$ is a countable subset of ...
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Factorization of permutations into two factors with fixed number of cycles, plus a placement condition

I have asked this question in MathOverflow, but it received no answers, so I am posting it here. In my recent work I have been led to consider the following type of permutation factorizations. Let ...
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1answer
23 views

How find the length of an array

Story: In fact this question is related to THIS. How to create an array maintaining following conditions- ...
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1answer
28 views

Principle of Counting [closed]

If you consider the numbers 1, 2, 3, 4 and 5, how many 3'digit numbers can formed; If repetition is not allowed and such that they are all: a.) greater than 200? b.) ending in 5? c.) odd numbers If ...
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2answers
20 views

Number of ways to distribute 4 different objects and 5 identical objects in 3 separate groups?

So, the question goes as: The number of ways in which 4 different toys and 5 identical marbles can be distributed between 3 different people, if each person gets at least one toy and one marble is? ...
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1answer
26 views

26 flavours of ice-cream, how many different banana splits can be made that have 3 different flavours?

A boutique ice cream bar stocks 26 flavours and offers a rainbow banana split that contains 3 scoops of ice cream, each of a different flavour. How many different rainbow splits can the store ...
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1answer
41 views

Permutation of letters, the principle of inclusion and exclusion [closed]

How many permutations of the letters ABCDEFG do not include ABCDE, EDAB, EDG, GFAB. My solution: $$7! - \left(\frac{7!}{5!} + 2 \cdot \frac{7!}{4!} + \frac{7!}{3!}\right)$$
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0answers
22 views

Permutation and Combination Problems [closed]

Permutation and Combination Problems Please Help me solve these problems :)
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2answers
36 views

In how many ways can 5 men and 5 women sit at a round table such that no 2 persons of the same gender sit next to each other?

In how many ways can 5 men and 5 women sit at a round table such that no 2 persons of the same gender sit next to each other? The book's answer is $2\times 5! \times 5!$ Why is it not $2\times 4! ...
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2answers
26 views

Counting ways to arrange the word REGULATIONS.

Find the number of ways the word REGULATIONS can be arranged such that there are exactly $4$ letters between $R$ and $E$ . I did $4!\ \ \ \ \text{for}\ \ ...
0
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2answers
26 views

Permutation and Combination Committee Questions [closed]

Q1. In how many ways can we select a committee of 6 persons from 6 boys and 3 girls, if at least two boys and two girls must be there in the committee? Given Answer: 65 Q2. In how many ways 7 persons ...
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2answers
28 views

How to calculate the minimum and maximum number of matches between two sequences?

I have two sequences of the same length $n=3$: $\{A,B,C\}$ and $\{A,A,B\}$. When I compare them, there is 1 match since both have an "$A$" in the first position. Generating all 6 permutated versions ...
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0answers
25 views

$S_n$ is isomorphic the permutations of the identity matrix?

Prove that the set of permutations $S_n$ is isomorphic to the group of invertible square matrices of order $n$ where each row has $n-1$ zeros and $1$ in one place. This is very intuitive to me, ...
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1answer
35 views

Permutation representations of symmetric group of small order

Thanks for any comments or help. What is the list of faithful permutation representations of $S_k$ of degree at most $n=2k$ for $k=3,4,5,6$? Is it possible to find its centralizer in each case?
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0answers
50 views

No. of ways to Generate the String [duplicate]

I want to generate a binary string, such that number of occurrence of $00,01,10$ and $11$ are to be fixed. How can we find out the numbers of ways for given value. For example: number of occurrence ...
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0answers
32 views

Find the different Binary String [duplicate]

I want to generate a binary String, such that number of occurrence of 01,10,00 and 11 are to be fixed. For Ex: Number of occurrence of 01,10,11 and 10 are 1 1 2 and 1 respectively. ...
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0answers
18 views

Unique unordered combinations of varying length

Given random set of integers. I.e. $\{1,2,2,3,3,3,5\}$ Find the number of unique, unordered of varying length sets that can be created. My Workings This is not a homework problem, but rather, ...
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1answer
63 views

find number of strings

Find the number of strings consisting of only a and b which have P occurrence of aa Q ...
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1answer
82 views

Number of binary numbers given constraints on consecutive elements

I've been trying to solve this question for quite a while, given to us by our discrete maths professor. I've been having a hard time in general with it, so I thought I tried looking it up online but ...
3
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1answer
44 views

Is it enough to determine if two finite groups are isomorphic if we can map the generators?

Heading: A General Inquiry about Finite Isomorphic Groups and Their Generators If I am given two finite groups whose generators I know, is it enough to show that by mapping the generators ...
2
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1answer
28 views

If $\alpha,\beta \in S_{n}$, and $\alpha\beta = \beta\alpha$, then $\beta$ permutes those elements left fixed by $\alpha$.

Here is my solution. Let ${a_1,...,a_k}$ represent all the integers that are permuted by $\alpha$, and let ${a_{k+1},...,a_{k+j}}$, where $k+j \leq n$, be all the elements that are left fixed by ...