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

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Find $T_1(\langle (1,2,3,4,5,6,7,8,9) \rangle )$

$T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ I am unsure what to do. Let that long permutation be $b$. Do we just find calculations of $b$ like $b^2$ or $b^{-1}$ or $b^b$ etc, that would give an ...
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2answers
37 views

How can I simplify this number theory problem?

Let X = {1, 2, 3, 4, 5, 6} and σ= \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 4 & 3 & 5 & 2 & 1 \\ \end{bmatrix} Define a relation ∼ on X as ...
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1answer
21 views

How do we solve these permutation and combination questions? [on hold]

Q1 In how many ways a panel of six doctors is selected from five surgeons and six physicians if condition is surgeons are more than physicians. A 82 B 81 C 65 D 135 Q2 Find the no. of ...
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1answer
453 views

binary strings and counting sequence problem

Hello i'm working on these questions and I have few questions 1) A binary string is a finite sequence of 0 and 1. Ex. 001101 is a string of length 6 a) List all binary strings of length 4 (so I ...
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1answer
26 views

Finding maximal product of numbers of permutations

Let $n\geq 1$ be a total number of objects that must be taken from $m\geq 1$ sets of objects. For all $i \in \{1,\cdots,m\}, \ M_i \in \mathbb{N}^*$ denotes the number of objects present in the set ...
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3answers
32 views

How to find the Permutation in S8 given as the product C= (1483)(12765)(34687)?

I don't understand how to answer this. Just by reading it off, "1 goes to 4, and 4 goes to 6, thus 1 goes to 6" but that logic doesn't match with the answer i've been giving: Answer: (127)(386)(45). ...
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9 views

Maximize the mutual permutation disparity

I am trying to work on a problem that needs me to find the top-k most disparate permutations for a n-tuple (hence n! possible choices). The disparity measure between two permutations I'm thinking of ...
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18 views

$A^{\pi}$ property

Can someone give an example for the matrix mentioned in the following definition. Definition is taken from "Iterative Solution Methods" of Owe Axelsson. Definition: The matrix $A$ said to have ...
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0answers
18 views

How do I calculate such possible number of total and serial schedule?

Consider the following two transactions $T_1$ and $T_2:$ How many non serial schedules are possible, if we execute both transactions concurrently? $3000$ $3001$ $3002$ $3003$ My try: ...
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1answer
29 views

Simply showing the addition of permutations

How can I show for example AB+BC+AC simply. It is adding up the permutations of n numbers. Another example would be ABC+ABD+ACD+BCD. Sorry I'll try to make it clear with an example ( which is sort of ...
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2answers
58 views

number of function $f$ from $f:\mathbb{A}\rightarrow \mathbb{A}$ and satisfying $f(f(x))=x$

Let $A=\{1,2,3,4\}\;,$ Then total number of function $f$ from $f:\mathbb{A}\rightarrow \mathbb{A}$ and satisfying $f(f(x))=x$ $\bf{My\; Try::}$ If $f(x)=x\;,$ Then $f(f(x))=x.$ So there are ...
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2answers
30 views

3 men have 4 coats , 5 waistcoats and 6 caps. Then in how many ways can they wear them?

The question is in the title itself. First, I would like to share how I solved this problem at first: We have $4$ coats, $5$ waistcoats and $6$ caps. So, I considered that each man wears one coat, ...
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2answers
35 views

100 shoelaces, pick 2 random ends and tie them together, what is the probability that a loop is created?

The question is: There are 100 shoelaces in a box. You pick two random ends and tie them together. Either this results in a longer shoelace (if the two ends came from different pieces), or it ...
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1answer
49 views

Characters of permutation representations for $S_4$

I am going through the lecture note How to get character tables of symmetric groups. On page 2, it computes the character table of $S_4$. The procedure starts with building the table of the ...
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1answer
14 views

Number of recursive permutations of all sizes

Consider you have a set of $n$ elements. Now, create all the possible permutations of $k$ elements. Finally, for each permutation create all the possible combinations with the permutations of the ...
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1answer
42 views

The number of Sylow $5$-subgroups in $S_6$

Find the number of sylow $5$-subgroups in $S_6$. First: $ord(S_6)=6!=2^4\cdot 3^2\cdot 5=144\cdot 5$, so $n_5|144$ and $n_5\equiv 1\pmod5$, where $n_5$ is the number of sylow $5$-subgroups. ...
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1answer
39 views

How many 4 digit pins on set {0-9}

A password can be any 4 digit {0...9}. 1.)How many possible passwords are there? for this I did $10^4 = 10,000$ 2.) How many possible passwords with no repeated digits? $10*9*8*7 = 5040$ 3.) How ...
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0answers
31 views

Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd S_\mathbb{N}$. [on hold]

Let $S_\infty \subset S_\mathbb{N}$ be the set of permutations of $\mathbb{N}$ which are the identity on all but a finite number of elements. Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd ...
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5answers
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A4 has no subgroup of order 6

Can a kind algebraist offer an improvement to this sketch of a proof? Show that $A_4$ has no subgroup of order 6. Note, $|A_4|= 4!/2 =12$. Suppose $A_4>H, |H|=6$. Then $|A_4/H| = ...
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2answers
29 views

4-Sequences {0…9}

My questions are given the set {0,1,2,3,4,5,6,7,8,9}, 1) How many 4-sequences are there? (would this be $10*10 * 10 * 10 = 10,000)? $ since the max possible numbers given to each 4 slots is 10. 2) ...
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1answer
25 views

5 letter password either lowercase or uppercase

Given that you can have 5 letter password that contains either lowercase or uppercase. My questions are: 1) How many possible passwords are there? I did $52^5 = 380,204,032$ since there are 52 ...
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0answers
21 views

Combination and Permutation How many words can be formed? [duplicate]

A contest consists of finding all code words that can be formed from the letters in the word "alpha".Assume that the letter "a" can be used twice but the others at most once: a)How many five-letter ...
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1answer
15 views

Product of permutations and subgroup generated by permutation(s)

I'm get confused while working with permutations, so I have some questions. $\sigma$ = (1,7,3)(2,10,4,8) $\rho$ = (3,7) $\tau$ = (1,7) First I am told to compute $\tau$$\sigma$$\tau^{-1}$ I dont ...
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2answers
69 views

Count the permutations which are products of exactly two disjoint cycles.

Let $a_n$ be the number of those permutation $\sigma $ on $\{1,2,...,n\}$ such that $\sigma $ is a product of exactly two disjoint cycles. Then find $a_4$ and $a_5$. Calculating $a_4$: Possible ...
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3answers
62 views

Number of permutations which are products of exactly two disjoint cycles. [duplicate]

Let $l_{n}$ denote the number of those permutations $f$ on the set $A=\{1,2,....,n\}$ such that $f$ is the product of exactly two disjoint cycles. Show that $l_{5}=50.$ I tried a lot but reached ...
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1answer
16 views

Odds of an event happening

Trying to get my head around the correct way of approach this. You are able to use any letter of the alphabet or number allowing for 36 options, with this you are to create PIN of length 4, for ...
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0answers
33 views

Levi-Civita symbol (permutation tensor)

I was going over a past exam and the following two questions came up: Show that the Levi-Civita symbol $\varepsilon_{ijk}$ is a tensor. Evauluate the following: ...
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1answer
31 views

Possible ranks of a $n!\times n$ matrix with permuted rows

Let $a_1,\cdots,a_n$ be $n$ arbitrary real numbers. Form the $n! \times n$ matrix $M$ whose rows are obtained by permuting the $n$ numbers given. Find all the possible ranks of such a matrix. ...
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49 views

Number of valid parenthesis

I have to find out the number of valid parenthesis.Parenthesis are of two type [] ,(). How many ways are there to construct a valid sequence using ...
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1answer
35 views

Number of possible subsequences

Given 4 integers - $A,B,C,D$ such that $A \leq B \leq C \leq D$ (i.e they are in non decreasing order). Now we need to find number of possible non decreasing subsequences $(W,X,Y,Z)$ such that $1 ...
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0answers
59 views

Twisty Puzzle Solving Program

I'm writing a program to help me solve a twisty puzzle. In this case it's the face-turning octahedron. I'm representing the puzzle as a group with face twists as generators. The facelets are in a list ...
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0answers
17 views

Number of Non Decreasing Sequence. [closed]

I have to find the number of non decreasing sequence (A,B,C,D) such that ...
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3answers
42 views

Seating arrangements of 7 boys and 5 girls in a row.

In how many ways can these boys and girls be arranged in a row if between two particular boys A and B there are no boys but exactly 3 girls?
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1answer
57 views

Decompose induced representation of $S_2$ and $S_3$

Let $ H=S_2 \subset G=S_3 $. Then use Frobenius reciprocity to decompose $ \operatorname{Ind}_H^G(\operatorname{sgn}_H) $ into irreducibles. $ G=S_3 $ has $ 3 $ irreps $ 1_G, ...
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35 views

Isomorphism of Non-Symmetric Matrix when Permutation-Set is given: A simple observation

Context: Consider, two $m \times n$ matrices $A, B$ such that there is a permutation $\kappa$ that such that such that $A^{\kappa}=B$ (Wielandt's notation), i.e. $A, B$ are isomorphic but not ...
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2answers
25 views

Proving that the index of a subgroup of $S_n$ that keeps a specific set invariant has a certain order

Let $n \in \mathbb{N}, n ≥ 2$, and $k \in \{1, 2, ..., n-1\}$, and let $A \subseteq \{1, 2, ..., n\}$ with $|A| = k$. Furthermore, let $G$ be a subgroup of $S_n$ that fixes $A$, i.e. for all $π \in ...
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2answers
39 views

Finding the Left and Right Cosets in $A_4$

I'm really struggling with a Group theory class and would love some help. HW Question is as follows. Consider the subgroups $H = \left<(123)\right>$ and $K=\left<(12),(34)\right>$ of ...
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3answers
68 views

Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
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1answer
26 views

Inversions of a permutation. Confused

Sorry for this basic question. In here, we have $2$ inversions of $1$ element (from the set $\lbrace 1,2,3\rbrace$): $$ 132, \\ 213, $$ and that $321$ is a $3$-element inversion permutation. Why ...
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1answer
630 views

$N^\text{th}$ (in lexicographical order) term of balanced brackets string

We have the following balanced brackets permutations of length $4\cdot2$ in lexicographical order: ...
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1answer
24 views
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1answer
20 views

Permutation test for equality between the distribution of $g$ population

I have a data matrix of 42000 observation and 12 variables I suppose to observe 12 samples of size $n_j$ from 12 indipendent random variables $Y_j,j=1,...,g$ I want do a permutation test for $$H_0: ...
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3answers
203 views

Product of permutation cycles, transpositions. Are there different conventions in the order?

From this answer I get that within each cycle you map each element to the one on the right, when taking the product of cycles the one on the right should be performed first, as a typical operator. ...
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0answers
28 views

Galois group of polynomials

Let $f$ be an irreducible polynomial over a field K, and $\deg f = 4$, with roots $a,b,c,d$. Let $g$ be a cubic resolvent with roots $\alpha,\beta,\gamma$. And $\alpha=ab+cd, \beta=ac+bd, ...
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1answer
30 views

Probability of selecting a jury

Does anyone know how to find the probability of selecting a jury of $12$ people ($6$ men and $6$ women) out of an initial group of $18$ people ($6$ men and $12$ women)? With my knowledge of ...
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0answers
35 views

12 numbered pigeonholes and balls [duplicate]

This problem was inspired by this James Randi challenge. Given $12$ numbered ($1$ to $12$) pigeonholes and $12$ numbered balls (also from $1$ to $12$); what is the probability that a random ...
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1answer
18 views

Total number of integral solutions to the factors of a given numbet

Let $a$ be a factor of $120$ then what are the total number of positive integral solutions to $xyz=a$ including 120. The answer is $320$ . After wasting almost $15$ mins in getting the factors of each ...
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1answer
26 views

Generating uniform permutations by a particular method

Let $A$ be a uniformly random permutation of the numbers $\{1,2,\cdots,n\}$. I want to generate a uniformly random permutation from $A$ on the numbers $\{1,2,\cdots,n,n+1,\cdots,n+m\}$. In other ...
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24 views

Number of ways to allocate balls

We are given $N$ buckets and $X$ ways to allocate balls in each possible pair of buckets. How many ways of distributing balls in all buckets exist ? For example - If we have $3$ buckets and $7$ ways ...
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1answer
22 views

Character of a representation on $S_3$ and irreducible representations

Here is the character table of S3: Consider $V=\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ with basis $e_{ijk} := e_i \otimes e_j \otimes e_k $ Let $\pi$ be the representation of ...