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

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24 views

Centralizer of $\sigma\in S_n$ [duplicate]

Let $\sigma\in S_n$ . Describe the centralizer of $\sigma$. Thought: If I conjugate $\sigma$, then $\tau^{-1}\sigma\tau=\sigma.$ This means that $\tau$ is a power of $\sigma,$ and $<\sigma>\...
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1answer
38 views

How many ways are there to arrange the letters of word $ALGEBRA$ such that the relative order of the vowels and consonants doesn't change?

I did this question this way :- there are 4 consonants in the words (LGBR) and there are 7 letters in the word. $therefore$ number of in which consonants can be arranged in relative order will be $C(...
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4answers
46 views

How many mixed double pairs can be made from 7 married couples provided that no husband and wife plays in a same set?

So for first man there can be 7 possible partners including his wife, for the next man there will be 6 possible partners and so on, therefore for $7$ men and $7$ women, there will be $7!$ possible ...
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3answers
80 views

Finding number of functions from a set to itself such that $f(f(x)) = x$

The questions states that $f: A\rightarrow A$ is a function which satisfies $f(f(x)) = x.$ We have to find the number of such functions with $A = \left\{1,2,3,4\right\}$. The given condition clearly ...
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2answers
40 views

A question of permutations and combinations with six cards and six envelopes.

Six cards and six envelopes are numbered 1,2,3,4,5,6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same ...
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2answers
34 views

How do I calculate the number of unique permutations in a list with repeated elements? [duplicate]

I know that I can get the number of permutations of items in a list without repetition using (n!) How would I calculate the number of unique permutations when a ...
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2answers
51 views

Counting the number of “distinct” permutations of two sets?

I don't really know how to introduce this question, so I start defining something I needed in order to well understand the problem I met! Let $A$, $B$ two finite sets of distinct elements, with $|A|=|...
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1answer
33 views

rationale for book's solution of combinatorics question about scheduling ten speakers with restrictions

If A, B, C are among $10$ people speaking at a function in alphabetical order What are total ways of doing so. BOOKS APPROACH: There are $10$ people out of which $3$ need to be taken care of. So ...
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1answer
39 views

Permutations of 3 digit numbers divisible by 5

I recently had to answer the following permutation question: How many 3 digit numbers can be formed from the digits 2,3,5,6,7,9 which are divisible by 5 and none of the digits are repeated? ...
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3answers
85 views

In how many ways can we arrange the letters of word BAHAMA such that it starts with H and ends with A?

In how many ways can we arrange the letters of word BAHAMA such that it starts with H and ends with A? I have a doubt in the selection of A at the last position. Please help. Thanks in advance!!
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2answers
144 views

In how many ways can the letters of word $PERMUTATIONS$ be arranged if there are always 4 letters between P and S?

In how many ways can the letters of word $PERMUTATIONS$ be arranged if there are always $4$ letters between $P$ and $S$? Now there are $12$ blank spaces, which we have to fill by the letters of the ...
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1answer
45 views

Combination and Permutation S= {A,B,B,C,C,C,D,D,D,D,E,E,E,E,E}.

S= {A,B,B,C,C,C,D,D,D,D,E,E,E,E,E}. If I choose n element from S, how many possible combination (unordered) and permutation (ordered) are possible (without using decision tree or counting)? What is ...
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1answer
37 views

Combinatorics Puzzle (Circular Table) [closed]

Q. Eleven members of a cricket team are numbered 1,2,3..11. In how many ways can they be seated around a circular table so that the numbers of any two adjacent players differ by one or two. (...
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2answers
78 views

Arrangement of letters in VISITING with no pairs of consecutive Is.

Could someone help me understand a book's solution to the following problem? I am providing my own solution, but I fail to understand theirs. Question: How many ways are there to arrange the ...
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2answers
72 views

Setting up an inclusion-exclusion question

What is the number of one-to-one functions $f$ from the set $\{1,2,...,n\}$ to the set $\{1,2,...,2n\}$ so that $f(x) \neq x$ and $f(x) \neq 2n - x + 1$ for all $x$? I'm getting that the number of ...
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0answers
11 views

Number of letters moved by a product of permutations

Let p and q be permutations in the symmetric group on n letters. p and q need not have the same cycle structure. Now compute q * inv(p) -- for inv(p) the inverse of p -- and count the number of ...
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1answer
14 views

Permutations starting with a specific letter

Ok, this is a homework question and I think I've resolved it but I want to bounce it off you guys. I have a $6$ letter word with no repeated letters. I need to calculate how many $3$ letter words can ...
2
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0answers
42 views

Positive integers $<100000$, how many contain exactly one $3$, one $4$ and one $5$

So I use $5$ positions for range $00000$ to $99999$ Choose $3$, choose $4$ and choose $5$ as follows: $5C1 \cdot 4C1 \cdot 3C1$ Remaining $2$ digits have $7$ possible digits as input Ans: $5C1 \...
3
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1answer
51 views

Painting a 2x2 Grid

We have a 2x2 grid and 10 different colours. I want to paint such that adjacent grids are painted with different colors. How many ways can i do this? ...
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2answers
31 views

$5$ chem students, $6$ maths students and $7$ physics students permutation

$5$ chem students, $6$ maths students and $7$ physics students. Find the number of arrangements if a)Chem majors are to occupy the first 5 positions b)Chem majors cannot occupy the first 5 positions ...
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0answers
11 views

Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph

I am trying to understand Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph. I quote it from Cycles in graphs and groups by Kantor. Theorem $1$ (...
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1answer
65 views

Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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2answers
38 views

Finding the parity of a permutation “exclusively”?

I'm trying to find the parity of permutations such as $(2468)$. What makes it possible to find the "exclusive" parity of such permutation? I.e. that if one tries to express $(2468)$ as a product of ...
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2answers
30 views

If $\sigma=(a_1 a_2 … a_n)$ and $|\sigma|$ is odd, then what is $\sigma^2$?

I'm trying to understand the way to infer the power of a permutation. If $\sigma=(a_1 a_2 ... a_n)$ is a $k$-cycle and $k=|\sigma|$ is odd, then how can I infer what $\sigma^2$ is?
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1answer
72 views

Prove $sgn(π) = sgn(π^{-1})$?

I'm pretty sure the inversion count of $π$ should be the opposite of the inversion count of $π^{-1}$. By this I mean if $π$ looks like this: $1 \to 1$, $2\to 2, \ldots, 10 \to 10$ and therefore the ...
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1answer
38 views

Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
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1answer
35 views

Write $π = (3, 2, 5)(2, 5, 4)$ in “table” notation?

Isn't this impossible...? Because this permutation goes from 3 --> 2 ---> 5 ---> 3 according to the first cycle, but goes from 2 --> 5 ---> 4 ---> 2 according to the second cycle. So 5 can't go to 3 ...
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1answer
28 views

Permutation of the alphabets of the word “mediterranean” such that first and fourth letter are “r” and “e” respectively.

Above is the original question. The correct answer is in green that is 59. I have chosen option 3 that is $\frac{11!}{(2!)^3}$ because I thought that there are 13 alphabets in the word "mediterranean"...
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1answer
50 views

Prove that the funtion f: $G\rightarrow G$, defined by $f(x)=x^k$, $x \in G$ is a permutation of $G$

Help me with this exercise, I could not do it :( Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove that the function f: $G \rightarrow G$, defined by $f(...
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17 views

Show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$

Here's the full question: If $σ ∈ S_n$ is any permutation and $i_1, . . . , i_k $ are $k$ distinct elements of $\{1, . . . , n\}$, show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , ...
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1answer
37 views

How can we compute restrictions from a character table?

I would like to how to, when given a character table, calculate the restriction. $Res_H^G : Rep(G) \rightarrow Rep(H)$. For example: Let $G=S_4$ whose character table is given below (see ...
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0answers
22 views

Need help calculating number of possible passwords with given criteria

I need help calculating the number of possible passwords with a given set of criteria. Here is the set of criteria: Passwords are case insensitive. Must be 6-14 characters. Must contain at least 1 ...
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1answer
17 views

Counting monomials with $k$ variables

Say we expand $\left(\sum_{i=1}^n x_i\right)^k$ into monomials. If $k=3$ there are $3n(n-1)$ monomials with two variables: $3x_1x_2^2 + 3x_1x_3^2 +\dots + 3x_1^2x_2 + \dots$. Is there a closed form ...
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2answers
54 views

Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$.

$S_n$ is the set of all permutations. Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$ I can somewhat see why this statement would be true, seeing as permutations are read from right to ...
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2answers
29 views

Suppose $π ∈ S_n$, and for this $π$ define $C_π : S_n → S_n$ be defined by $C_π(σ) = πσ$. Why is $C_π$ a bijection?

$S_n$ is the set of all permutations. I'm just starting on this material, so I'm confused on how to read this problem. Does the function consist of multiple permutations (i.e. the permutation of a ...
2
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1answer
24 views

Find The Number Of Outcomes

I understand how to find the number of outcomes using permutations and combinations, but then I thought to myself what happens when it involves both? I will make a mock scenario to explain what I am ...
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0answers
22 views

the length of the conjugate class containing $\alpha$ in $S_n$ [duplicate]

Suppose $\alpha$ $\in$ $S_n$ and there are exactly $n_i$ $l_i$-cycles ($i=1,2, ... ,k)$ (containing $1$-cycles) in the cycle decompostion of $\alpha$ ., then the length of the conjugate class ...
2
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1answer
21 views

Converting Permutations to Combinations: Simple Stats in Practise

In a popular text book there is a question that has bothered me that I am sure is very simple for others and I'm just missing something..... So image $100$ songs and we have $10$ as Beatles songs. We ...
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1answer
22 views

Generate a unique combination from an index within the number of combinations

I'm writing a program which will use a genetic algorithm optimize neural networks to play tic-tac-toe (That's not related), and I've come across the following problem: I'm looping through every ...
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1answer
20 views

Permutation : Is there any formula to solve this?

Given, 14 objects of type A 8 objects of type B 3 objects of type C 2 objects of type D Find the permutation of 10 objects? Is there any general formula in permutation to solve a problem ...
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2answers
42 views

What is an intuitive explanation of the combinations formula?

I perfectly understand the permutations formula i.e. if you have $n$ things how many ways can you rearrange it if taken $k$ at a time (or if you have $k$ slots)? So you draw the following tree. And ...
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0answers
27 views

Is $N_{A_7}(H) = H$, with the following $H$?

I am following a proof in which I have a subgroup of $S_7$ defined by $H := \langle (2, 3, 4)(5, 6, 7) , (2, 7, 6, 3)(4, 5) \rangle$ The book implicitly uses that $N_{A_7}(H) = H$ (the normalizer ...
2
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1answer
50 views

About conjugating a $7$-cycle in a subgroup of $S_7$

Following a proof in which I have a transtive group $G$ of order $168$ , which is a subgroup of $S_7$ (I am trying to characterize it, I cannot use well know facts such as it is always isomorphic to $...
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1answer
43 views

Proof to an observation of stabilisers and orbits

Observation: If $\alpha^{g}=\beta$ then $G_{\beta}=g^{-1}G_{\alpha}g$ Just to get the notation out of the way: $G_{\beta}= g^{-1}G_{\alpha}g$ is the stabiliser of a point element $\beta$ in a ...
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1answer
44 views

Number of Elements in a Conjugacy Class of $S_N$ (Derivation)

Consider the conjugacy classes of the symmetric group $S_N$. Each conjugacy class consists of permutations that have the same cycle structure. We see that the number of possible cycle structures is ...
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2answers
41 views

An automorphism that has no fixed points except for the identity and is its own inverse implies commutativity

Let $G$ be a finite group and suppose there exists $f\in\text{Aut}(G)$ such that $f^2=\text{id}_G$, i.e., $f$ is its own inverse, and such that $f$ has no fixed points other than the identity $e$ of $...
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0answers
12 views

Removing the dimension factor in Fannes inequality

Given two distributions $x=(x_1,\ldots, x_n),y=(y_1,\ldots y_n)$ on $[n]$, it is known by Fannes inequality that $H(x)-H(y)\leq O(\|x-y\|_1\log n)$, where $H(\cdot)$ and $\|\cdot\|_1$ represent ...
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0answers
24 views

Derangements One to One Functions

What is the number of one to one functions from $\{1,2,3,\cdots,n\}$ to $\{1,2,3,\cdots,n\}$ so that $f(x)\neq x$ for all $x$. I understand that $A_1 \cup A_2 \cup A_3 \cup\cdots\cup A_N$ is the set ...
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2answers
31 views

Decomposition of permutation

I was asked to decompose the permutation $$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1 \\ \end{pmatrix} = (12345) \in S_5$$ into a product of two ...
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1answer
25 views

Polynomial roots conditions vary with coefficients

Polynomial equation $\sum_{i=0}^4 p_i x^i=0$ have the following root conditions: 1) $a_0 \pm b_0 i, a_1 \pm b_1i$ 2) $a_0 \pm b_0 i, a_1, a_2$ 3) $a_0, a_1, a_2 \pm b_2i$ 4) $a_0, a_1, a_2, a_3$ I'm ...