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

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8 views

Possible number of sequences

How many finite sequences are there such that $x_{i} = 1$ or $ 2$ and $\sum_1^n x_{i} = 10$ ? Now I did it this way: number of $1$'s $\ $:$\ $ $0$ ,$2$ ,$4$ ,$6$ , $8$ , $10$ and ...
2
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1answer
17 views

Permutations of {1, 2 .. 30} where $a_n - a_n-m$ is divisible by m from {2, 3, 5}

There are $N$ permutations $(a_1,a_2,\dots,a_{30})$ of $1,2,\dots,30$ such that for $m\in\{2,3,5\}$, $m$ divides $a_{n+m}-a_n$ for all integers $n$ with $1\leq n <n+m\leq 30$. Find $N$. I really ...
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1answer
20 views

How many structurally different latin squares of order 5 do exist?

I know the number of latin squares order 5 which start with 1 2 3 4 5 in the 1st row or column, that is 1344, but the greater part of that number consists of structural duplicates of each other. So, I ...
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1answer
69 views

If $g$ is a permutation, then what does $g(12)$ mean?

In Martin Lieback's book 'A Concise Introduction to Pure Mathematics', he posts an exercise(page 177,Q5): Prove that exactly half of the $n!$ permutations in $S_n$ are even. (Hint: Show that ...
3
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2answers
42 views

Permutation count of AABBC

Given a string: $AABBC=A^2B^2C^1$ I am trying to find the Total Permutations (this may be incorrect): $\dfrac{5!}{2!\cdot2!}=30$ My question is how would I find the partial sums (perhaps the ...
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2answers
56 views

How many two letter words can be formed from 26 English letters?

There are 26 English letters. From layman approach, How can one calculate the possible two letter words from these 26 English letters?
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2answers
34 views

Elements of $S_n$ which can not be product of $\leq n-2$ transpositions

It is well known that every element of $S_n$ can be written as a product of at most $n-1$ transpositions. This theorem is proved in all the books which discuss the permutation groups. But, I find that ...
3
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2answers
37 views

How to conceptualize “dividing out” a number (e.g. in permutations, Bayes' Theorem)?

I'm trying to achieve a better conception of what it means to "divide out" a variable/number, because I'm currently have a lot of trouble justifying to myself why it actually works the way it does in ...
0
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3answers
27 views

Find the number of seven digit whole numbers in which only 2 and 3 are present as digits if no two 2's are consecutive in any number?

Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number? My Approach: We can make numbers and see like: ...
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2answers
57 views

How many permutations of the word TOMORROW can be made if the O's can't be together?

I'm trying to answer this question. This is my attempt of solution: First we distiguish the O's and R's, then we have the word: $TO_1MO_2R_1R_2O_3W$. We have $8!-7!\cdot3!-6!\cdot 3!$ different ...
3
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3answers
35 views

Combinatorics Question - Permutations and Supersets

I had a question that seems pretty straightforward, but I can't seem to wrap my mind around it. Let's say I have a bunch of elements in a set. {A, B, C, D, E}. How many permutations are there of ...
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0answers
51 views

Permutation of numbers from multiple sets [May contain duplicate numbers among other sets], resulting in Non-Duplicate Set

We have 3 Data Sets. From each set we will be selecting few numbers. 3 from Set 1, 2 from Set 2, 3 from Set 3. Totally, we will get 8 Numbers from 3 Sets. The resulting sets shouldn't contain any ...
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3answers
51 views

Why is it true that $(a_1 \ a_2 \dots a_r) = (a_1 \ a_r)(a_1 \ a_{r-1})\dots(a_1 \ a_3)(a_1 \ a_2)$?

In the theory of permutation, a $r$-cycle $(a_1 a_2...a_r)$ is defined in the following way: Start from $a_i$, a permutation function $f$ sends $a_i$ to $a_{i+1}$. When $i=r, a_i \text{ will be ...
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2answers
54 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.
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2answers
37 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...
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1answer
28 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
31 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)?
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0answers
62 views

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

[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 ...
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2answers
48 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 ...
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1answer
22 views

Probability of a user references in a network [closed]

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
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1answer
53 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 ...
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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
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1answer
29 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 $A$. ...
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0answers
32 views

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

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 ...
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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
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0answers
52 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
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1answer
36 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 ...
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3answers
95 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 ...
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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 ...
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0answers
91 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
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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
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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 ...
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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.
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2answers
248 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
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1answer
80 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 ...
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2answers
114 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 ...
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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 ...
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4answers
48 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 ...
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0answers
34 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 $21^3$ ...
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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 ...
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4answers
111 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 ...
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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 $\{1,2,...,n\}$ ...
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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
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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 ...
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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 ...
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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 .
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5answers
94 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
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1answer
27 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 ...
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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} ...