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

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1
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3answers
44 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
0
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2answers
36 views

Number of permutations of an integer

If $n$ is an integer, how many permutations are less than, equal to and greater than $n$? For example if $n=24335$, $43325\gt n$, $23345\lt n$, etc...
1
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1answer
23 views

Functions that are “balanced” on the support of a permutation

Let $F = GF(2^n)$. Let $P(x), Q(x) \in F[x]$ be such that $P(x)$ is a permutation, while $Q(x)$ is not a permutation. For $\lambda \in F^*$ define the function $g_\lambda(x) = Tr(\lambda Q(x))$. Let ...
0
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2answers
28 views

Permutational Question

When I use the equation $P = \frac{n!}{(n - r)!}$ with n = 3 and r = 2, I get 6 permutations. Though, how do I get the amount of permutations without cross-duplicates(e.g A/B and B/A)?
0
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0answers
53 views

Any hint on : Every $A_{n}$ elemnt is $n$-cycles product. [on hold]

[Added explanation] I found this exercise as follows in Hungerford : Abstract algebra (3rd edition) page 236, exercise number 40. Stated as follows : C.40. Prove that every element of $A_{n}$ is ...
2
votes
2answers
44 views

Permutations: Interpreting Image Notation

I have a problem in interpreting permutation. I think the definition and my interpretation of it don't match each other. Let $\sigma=(1\ 2\ 4\ 3)$, and $\tau=(1\ 3\ 2\ 4)$ in one-line notation. I ...
-2
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1answer
22 views

Probability of a user references in a network [on hold]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
2
votes
1answer
50 views

NP combination puzzle (Klotski)

I've written a C++ program to solve sliding puzzles games such as UnblockMe and Car Parking. I'm quite happy about it, since it solves various schemes in less than a second. Recently I fed the game ...
1
vote
0answers
25 views

How many permutations cover alternating/reverse alternating permutations?

Given integers $1$ through $2n$, let $S$ be set of ordering of integers that respect even alternating or reverse alternating permutations (https://en.wikipedia.org/wiki/Alternating_permutation) up to ...
2
votes
1answer
28 views

Where do I use that $G$ is a permuation group?

This is about question $4.1.7$ from Dummit and Foote, and also related to my previous question. The question is (summarised a bit): Let $G$ be a transitive permutation group on a finite set $A$. ...
-3
votes
0answers
32 views

How to prove the theorem on group algebra of permutation group [on hold]

How to prove the following theorem: If $t$ is a vector in group algebra of permutation group, then $\cal{y}t\cal{y}=\lambda_t \cal{y}$, where $\cal{}y$ is the Young operator of permutation group and ...
0
votes
1answer
43 views

Compose $(1243)$ and $(5)$

Checking my work. In either direction: $(1243)[1] = 2$ and $(5)[2] = 2$, so far we have $(1, 2,\ldots$ $(1243)[2] = 4$ and $(5)[4] = 4$, so far we have $(1, 2, 4,\ldots$ $(1243)[4] = 3$ and ...
0
votes
0answers
48 views

Finding permutation matrix

Let $P_{\pi}$ denote a permutation matrix associated to the permutation $\pi:\{1,...,n\}\rightarrow \{1,...,n\}$ and $\sigma$ denote the cyclic permutation $(1 2 ...n)$. If T is the $n\times n$ lower ...
0
votes
1answer
35 views

How many permutations $(a_i)_{i=1}^{30}$ of $\{1,\dots,30\}$ satisfy $m$ divides $a_{n+m}-a_n$ when $m \in \{2,3,5\}$ and $1 \le n<n+m \le 30$?

Define a permutation $(a_1,a_2,\dots,a_{30})$ of $\{1,2,\ldots,30\}$ as good if for all $m \in \{2,3,5\}$, we have that $m$ divides $a_{n+m}-a_n$ for all integers $n$ satisfying $1 \leq n < n+m ...
4
votes
3answers
94 views

Why Composition and Dihedral Group have reverse order of operation?

NOTE - I didn't receive any answer in here and I think because my first post is not clear, so I entirely made another example: $K={\{id,r^2,r^4,s,r^2s,r^4s}\}$ is a proper subgroup of the dihedral ...
1
vote
1answer
76 views

Intuitively and Mathematically Understanding the Order of Actions in Permutation GP vs in Dihereal GP

I define $r$ to be one rotation clockwise, and s to be reflection on the 'horizontal' line (see the figure). So I can make these bijections: (in clockwise order) $$\begin{align*} 1,2,3,4,5,6 ...
0
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0answers
65 views

Permutation equivalence classes with kendall-tau distance

Consider a set $A=\{a_1,...,a_m\}\subset \{1,...,n\}$ for which $a_i<a_{i+1}$ for all $i = 1,\ldots,m-1$. Take any two distinct permutations $\sigma, \tau$ of $\{1,...,n\}$ such that $ ...
0
votes
1answer
32 views

Permutation and signature matrices “almost commute”

Let $\mathcal{P}$ be the set of all permutation matrices of order $n$ and $\mathcal{S}$ the set of all signature matrices of order $n$. Furthermore, let $$\mathcal{P}\mathcal{S} = \{PS \mid ...
3
votes
1answer
50 views

Characterisation of the squares of the symmetric group

I found out that for $n\le 4$ we have $S_n^2=A_n$ with $G^2$ defined by $$G^2:=\{g^2 \mid g\in G\}$$ for any group $G$. Surely we have $S_n^2\subseteq A_n$ for all $n\in\mathbb N$. Is there a ...
-5
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0answers
31 views

Permutations 123456 [closed]

User passwords for a certain computer network consists of 3 letters followed by 3 digits . How many different passwords are possible? Repetition is allowed.
11
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2answers
246 views

Show that there is always a way to achieve det(A) > 0

a) Assume that $(a_1, ..., a_9)$ are different positive numbers. Let us make a 3x3 matrix $A_s$ by placing them arbitrarily into 9 positions available. Show that there is always a way to assemble ...
2
votes
1answer
79 views

Proving a certain lemma about subgroups of $A_n$

In proving $A_n$ is simple for $n\neq4$, my teacher established the cases 1, 2, 3 as obvious, then proved the case 5, and proceded by induction on the rest. In the midst of that induction, he stated ...
3
votes
2answers
96 views

How to arrange 15 women and 15 men so no two females are seated next to each other?

To a certain conference, each firm can send two employee representatives, on the condition that one of them is a male and the other a female. If 15 firms were represented in this conference, what is ...
2
votes
4answers
97 views

When will Andrea arrive before Bert?

The question was as follows- on any given day, Andrea is equally likely to clock in at work any time from 8:50am to 9:06am. Similarly, Bert is equally likely to to clock in at work at any time ...
1
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4answers
47 views

Prove that the 4-group V is normal subgroup of $S_4$ by using isomorphism theorem

Prove that the 4-group V is normal subgroup of $S_4$ First, by using the multiplication table, I am able to prove that 4-group V is subgroup of $S_4$. But I face problem in proving that $\forall ...
1
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0answers
33 views

Simple string permutations question

How many sequences of 5 letters are there in which exactly two are vowels? My approach There are $5^2$ different permutations for 2 vowels and $\binom{5}{2}$ ways allocate them. There are $21^3$ ...
1
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2answers
40 views

Basic Password permutation question

I'm reading the problem from this stanford material (http://infolab.stanford.edu/~ullman/focs/ch04.pdf). Can you please help me understand this? Question: At Real Security, Inc., computer passwords ...
1
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4answers
96 views

How many words can be formed using all the letters of “DAUGHTER” so that vowels always come together?

How many words can be formed using all the letters of "DAUGHTER" so that vowels always come together? I understood that there are 6 letters if we consider "AUE" as a single letter and answer would be ...
0
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1answer
36 views

$S_n$ notation in permutations

What does notation $S_n$ stands for? For example if I have the following set $\{1,2,3,4\}$ so we say that $S_4$=24? Moreover in many examples I saw the use of following numbers like $\{1,2,...,n\}$ ...
1
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4answers
34 views

Question on Permutations Please advise

Among all seven digit decimal numbers,how many of then contain exactly three 9's? My Approach: 3 places contains only 9's---> 1*1*1 (No. of Ways to Choose out of 0 to 9) other 4 places: since first ...
3
votes
1answer
30 views

Which permutation am I? Or: what is a bijection $f:S_n \rightarrow \{1,2,\ldots,n!\}$ such that we can compute $f(\beta)$ easily?

Let $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$ and assume that $S_n$ is ordered in some way, i.e., $$S_n=\{\alpha_1,\alpha_2,\ldots,\alpha_{n!}\}.$$ We are able to choose this ordering on ...
0
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1answer
35 views

Can one modify the generators of a transitive group to get an intransitive group while preserving conjugacy classes?

There is a general question I'm interested in: given $g$ and $h$ with $H=\langle g,h \rangle$ a transitive subgroup of $S_n$, when is it possible to find $g',h'$ so that $H'=\langle g',h' \rangle$ is ...
0
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2answers
21 views

Commutativity of cycles

Disjoint cycles commute: $(ab)(cd) = (cd)(ab)$, but do non-disjoint cycles commute? Does $(ac)(ab) = (ab)(ac)?$ Consider the composition of two permutations: $\begin{pmatrix} a & c\\ ...
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2answers
31 views

The number of times the digit 8 will be written when listing the integers from 1 to 1000 [closed]

When i calculated the answer as 360 but in book it is mentioned 300. Please Help .
0
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5answers
92 views

Help needed to solve combinatorics problem.

I have been revisiting my old probability courses and I found a problem, which I can't figure out how to solve or at least what I get differs from the answer in the book. The problem reads as ...
0
votes
1answer
22 views

Number of permutations in a word ignoring the consecutive repeated characters

Given a word "aab", permutations are: aab, aab, aba, aba, baa, baa I need to get the number of permutations where characters don't repeat. So from the above permutations, I need to ignore those ...
0
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0answers
33 views

Trying to learn how to compose permutations

I am trying to prove myself that $(1)(2)(3)(4) = (12)(12)(3)(4).$ So, $\begin{pmatrix} 1 & 2 \\ 2 & 1 \\ \end{pmatrix}$ $\begin{pmatrix} ...
1
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1answer
59 views

Defeating enemy crab by cutting off legs and claws [closed]

The following is from the MIT-Harvard Tournament: You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and ...
0
votes
0answers
26 views

Problem with random permutation and conditional probability

Let $\pi_1,...,\pi_n$ be a random permutation of numbers $1,...,n$. If you are told that $\pi_k > \pi_1,...,\pi_k > \pi_{k-1}$, what is the probability that $\pi_k = n$? What I've tried: Let ...
2
votes
2answers
28 views

Permutations with Condition

I have looked at this old problem in my textbook: How many permutations $\pi \in S_n (n \geq 3)$ meet the requirement: $\pi (1) < \pi (2) $ or $\pi (1) < \pi (3)$? I am not sure how to ...
0
votes
0answers
62 views

Examples of injective maps such as MTF (Book Stack). Set of such mappings.

Let $S \in \mathbb N$. Let $\mathfrak S_S$ be the set of permutations of size $S$. Consider map $f : \mathfrak S_S \times \{1,2,\ldots,S\} \to \mathfrak S_S$, such as $f(\alpha, \cdot) : ...
0
votes
0answers
37 views

For which integers $r$ is $\sigma ^r$ also a $k$-cycle? [duplicate]

Let $\sigma$ be a $k$-cycle in $S_n$. For which integers $r$, is $\sigma ^r$ also a $k$-cycle? I think I managed to prove that this is true iff $(k,r)=1$, but my proof was too long and not elegant ...
2
votes
1answer
86 views

Rank 3 permutation groups

Let $G \leq Sym(\Omega)$ be a finite permutation group of rank 3, $\alpha \in \Omega$ and $g,h \in G$ such that $x_1 := g(\alpha)$ and $x_2 := h(\alpha)$ are not equal. Now my question is: Is there ...
0
votes
1answer
18 views

Hashing: Quadratic Probing

I have the following to prove, unfortunately I am not able to do so. Let h, h' be hash functions: $h(k,i) = (h'(k) + c_{1}i + c_{2}i^2)$ mod $m$. Show the following: if m is prime and $c_{2} \neq 0$ ...
1
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1answer
28 views

What is the distribution of cycle lengths in derangements? In particular, expected longest cycle.

There is a lot of information about expected cycle lengths in random permutations, but I'm having trouble adapting the arguments and calculations to the specific case of derangements - permutations in ...
1
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2answers
27 views

To prove an identity in permutation and combination.

I am trying to prove the following identity: ${n \choose 0}$ + ${n \choose 1}$ + $\ldots$ + $\frac{1}{2}{n \choose n/2}$ = $2^{n-1}$ where $n$ is even I know that I have to use few relations like ...
2
votes
1answer
36 views

Disjoint Cycles in a Cyclic Subgroup of $S_n$

If a permutation $\sigma$ $\in$ $S_n$, the permutation group of n elements, and $\sigma$ can be expressed as a product of disjoint cycles, is it necessary that the disjoint cycles be elements in ...
1
vote
3answers
64 views

In how many ways can a natural number be written as a sum of $2$ natural numbers?

For example, $7=1+6,2+5,3+4$. Hence $7$ can be written as a sum of $2$ natural numbers in $3$ ways.
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0answers
35 views

12- In a standard deck of 52 cards, how many ways can you deal out 4 cards that are all black or all not face cards? [duplicate]

I did the sad mistake of taking math in summer school to boost my average. I am stuck on a few questions. In a standard deck of $52$ cards, how many ways can you deal out $4$ cards that are all ...
0
votes
1answer
54 views

Random permutation and isolated points on the line

Let $[n]=\{1,\dots,n\}$ be the (ordered) set of the $n$ first integers, and $\mathcal{S}_n$ denote the set of permutations of $[n]$. Let $1\leq k \leq \frac{n}{4}$ be an integer. If I draw uniformly ...