questions regarding the re-orderings of some finite set of objects.
3
votes
1answer
44 views
Action of $S_7$ on the set of $3$-subsets of $\Omega$
Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem:
Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
1
vote
1answer
41 views
Permutation combination problem
This is how Edward’s Lotteries work. First, 9 different numbers are selected. Tickets with exactly 6
of the 9 numbers randomly selected are printed such that no two tickets have the same set of ...
1
vote
2answers
66 views
Product of permutation matrices
I want to prove that the product of two permutation matrices is itself a permutation matrix. But I don't know how. Please help!
2
votes
1answer
28 views
What is the professional term for the combination of the selection in n out of the total m elements?
I know the number of combinations is called ${}_nC_r$, but what about all the exact outcomes?
For example: I have $3$ elements $a,b,c$ and for the parameter $2$, I will have outcomes
$$ab,\quad ...
0
votes
1answer
24 views
Combination of arrangement and probability
Four guys and four girls are arranged in a row such that no two girls are together. What is the probability that any two of the four guys are together?
1
vote
1answer
24 views
possible combinations of 3-digit
How many possible combinations can a 3-digit safe code have?
Because there are 10 digits and we have to choice 3 digits from this,
then we may get $10^P3$ but A author used the formula $n^r$, why is ...
0
votes
3answers
30 views
permutation/combination problem
There are 3 doors to a lecture room. In how many ways can a lecturer enter the room from one door and leave from another door?
I have done like this: They way of entering is 3 and exiting is also
...
5
votes
1answer
58 views
Is there a name for this given type of matrix?
Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$?
(The motivation for this ...
2
votes
2answers
22 views
Calculating permutations if the sequences have to be in ascending order?
How would you go about calculating the number of permutations in ascending order.
Obviously if you had (a set of) 3 numbers you have $ 3! $ permutations:
(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), ...
6
votes
1answer
39 views
Grouping natural numbers into arithmetic progression
I need to find the number of ways of dividing the first 12 natural numbers into 3 equal groups (4 numbers each), so that the numbers in any particular group can be arranged in AP (Arithmetic ...
1
vote
3answers
57 views
The number of words that can be made by permuting the letters of _MATHEMATICS_ is
The number of words that can be made by permuting the letters of MATHEMATICS is
$1) 5040$
$2) 11!$
$3) 8!$
$4) 4989600$
First of all I do not understand the statement of the problem, I would ...
-2
votes
1answer
58 views
Show that there does not exist a permutation α in S4 satisfying (1 2)(3 4)α = α(1 2 3 4)? [closed]
Let $f=(1 8 )(3 6 4)(5 7)(2)$ and $g=(1 2 3)(4 5)(6 7 8)$ find signature of $f$. if there is a $α$ in $S_8$ such that $αf= gα$. Give reason for your answer.
2
votes
1answer
22 views
Number of Total orders of a dependency graph
Define a dependency graph to be a graph $G=(V,E)$ such that an edge between vertices $v$ and $u$ in $V$ is present if $v<u$ i.e. $v$ comes before $u$ in our ordering (I'm not very concise here, I ...
1
vote
1answer
21 views
Is it possible to divide n in d parts where each part can have a specific range of values?
Given a value n and number of parts d.
Each part has max and min values it can have.
Is it possible to divide n in d parts fullfilling the max and min value criteria
?
Example
d=2 n=5
Part 1 ...
1
vote
2answers
64 views
Permutations of a queue of interlaced boys and girls.
Suppose $5$ boys and $4$ girls are to be arranged in a queue such that between any
two boys there is at least one girl. Find the number of such arrangements possible.
What i think is $5$ boys ...
2
votes
6answers
50 views
calculate the number of possible number of words
If one word can be at most 63 characters long. It can be combination of :
letters from a to z
numbers from 0 to 9
hyphen - but only if not in the first or the last character of the word
I'm trying ...
0
votes
0answers
57 views
Ball and holder problem [duplicate]
I am trying to solve this but having a tough time deriving the formula.
There are $X$ ball and $Y$ holders $Y \leq X$. Out of the $X$ balls, $N$ are red and $X-N$ are blue.
What is the probability ...
1
vote
2answers
31 views
Number of length-five words
How many length-five words can be written using two A's, two T's and one E?
Why is it not $\binom5 2 \times\binom 5 2 \times \binom 5 1$?
Is it $ \binom 5 3 = 10$?
1
vote
1answer
29 views
Need an algorithm to compute number of elements in sample space
An urn contains $X$ red balls, $Y$ green balls, and $Z$ white balls. $N$ balls
are drawn without replacement from the urn, and the colors are noted in sequence.
$N \leq X+Y+Z$
Trying to come up ...
1
vote
0answers
36 views
Proving $Perm(S)\cong Perm(T)$
I am trying to prove this proposition:
Let $f:S\to T$ be a bijection of sets. Then $Perm(S)\cong Perm(T)$.
I clearly want to show that $\phi: Perm(S) \to Perm (T)$ is an isomorphism with, $\sigma ...
0
votes
0answers
31 views
How do I make a function where the range of the function is a permutation of the given function domain?
An example for 0, 1, 2, 3, 4, 5 would be:
f(0)=5;
f(1)=1;
f(2)=0;
f(3)=4;
f(4)=2;
f(5)=3;
I have found f(x) = 911 * x % N to work where 911 can be any large ...
1
vote
1answer
23 views
Dice Roll Permutation Problem
Here is my problem:
You have a standard dice, with possible rolls: $\{1, 2, 3, 4, 5, 6\}$. How many permutations exist in 10 rolls such that no two immediate rolls are the same?
For example:
$\{1, ...
2
votes
2answers
35 views
From Combination to Permutation
I am facing a (probably) basic counting issue.
If $P(n,r)$ the permutations for $r$ objects from $n$ and $C(n,r)$ the combinations, we have : $P(n,r) = r!C(n,r)$.
Yet there are two example in which ...
8
votes
0answers
58 views
Expression of basis vectors of permutation modules in different bases.
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
3
votes
4answers
96 views
What is the order of $\tau = (5\ 6\ 7\ 8\ 9)(3\ 4\ 5\ 6)(2\ 3\ 4\ )(1\ 2)$? Is $\tau$ an even or an odd permutation?
In $S_9$, what is the order of $\tau = (5\ 6\ 7\ 8\ 9)(3\ 4\ 5\ 6)(2\ 3\ 4)(1\ 2)$? Is $\tau$ an even or an odd permutation?
For the first question: I tried to write $\tau$ as the composition of ...
1
vote
1answer
58 views
Permutation & Combination - how many numbers smaller than $2.10^8$ and are divisible by $3$ can be written by means of the digits $0$,$1$ and $2$
How many numbers smaller than $2.10^8$ and divisible by $3$ can be written by means of the digits $0$,$1$ and $2$?
Left Zero padding not allowed.
I am getting this as -
3 digits - 2*3 = 6
4 digits - ...
2
votes
1answer
44 views
Permutation & Combination - How many 4 digit no's are there whose decimal notation contains not more than two distinct digits?
How many 4 digit no's are there whose decimal notation contains not more than two distinct digits?
1
vote
3answers
61 views
The number of bijections $f$ of $\{1, 2,…, n\}$ such that $f(i) \ne i$ for any $i$
Show that the number of bijections $f$ of $\{1, 2,..., n\}$ such that
$f(i) \ne i$ for any $i$ is equal to $$\sum_{j=0}^{n}(-1)^j\frac{n!}{j!}.$$
Can I get some help for the above problem? I am ...
1
vote
2answers
87 views
Can you find an isomorphic group?
Let $G$ be a group with elements $\{e, a, b, c, \theta, \theta a, \theta b, \theta c \}$ where $a^2 = b^2 = c^2 = \theta$, $\theta^2 = e$, $ab = \theta b a = c$, $bc = \theta c b = a$, $ca = \theta a ...
2
votes
2answers
57 views
At least two people have the same birthday
If there are 85 students in a statistics class and we assume that there are 365 days in a year, what is the probability that at least two students in the class have the same birthday?
I tried solving ...
1
vote
1answer
27 views
On permutations and Combinations
$mn$ squares of equal size are arranged to forma a rectangle of dimension $m$ by $n$, where $m$ and $n$ are natural numbers.
Two squares will be called 'neighbours' if they have exactly one common ...
3
votes
1answer
52 views
How many ways to build a hamburger?
Friendly's recent advertising campaign claims there are over 10 trillion combinations of hamburgers. Their options are:
Protein (4)
Bread (5)
Cheese (7)
Premium topping (3)
Hot toppings (10)
Cold ...
2
votes
2answers
30 views
Variance of derangements
Suppose I choose a random permutation on n numbers. It is easy to prove that the mean of the number of fixed points (i.e. the numbers that get mapped to themselves) is 1. Is there an easy (constant) ...
3
votes
2answers
40 views
Unable to get to all permutations after $n-1$ transpositions
Problem: Give an example of a permutation of the first $n$ natural numbers from which it is impossible to get to the standard permutation $1,2,\ldots,n$ after less than $n-1$ transposition operations ...
0
votes
3answers
33 views
Notation: permutation and its inverse
Consider the sequence $S = (A, B, C, D, E)$ and the permutation $\pi = (4, 1, 3, 5, 2)$:
Which of the following is true?
$$ \pi(S) = (B, E, C, A, D) \quad and \quad \pi^{-1}(S) = (D, A, C, E, B) ...
2
votes
2answers
44 views
Number of rectangles with odd side lengths on a chess board?
Given an 8x8 chess board, how do we find the total number of rectangles with odd side lengths?
(Both sides have odd length).
In general, what would be an elegant method to deal with problems like ...
0
votes
2answers
34 views
How to count permutations with restrictions on how items are grouped
I am trying to solve the following problem:
A town contains $4$ people who repair televisions. If $4$ sets break down, what is the probability that exactly $i$ of the repairers are called? Solve ...
0
votes
1answer
42 views
Counting Methods: Restricted Permutations
I have been scratching my head for a long time. The question is: How many words can be formed using all letters in the word EXAMINATION in such a way that the first two letters are different ...
1
vote
0answers
27 views
Continuous random sampling with replacement.
Construct a set $s\subseteq[0,1]$ by sampling points in $[0,1]$ with uniform probability density $x\leq1$ so that $|s|=x$. Interpret this as a sampling frame during which data is captured. Now, ...
0
votes
1answer
23 views
Plausible gene sequences
I'm not looking for a specific answer to a question (below). I think it is likely that the 'kind' of problem I have has been studied (and has a name ;). But I don't know what that might be. So I'm ...
0
votes
2answers
48 views
How many ways are there to encode the 26-letter English Alphabet into 8-bit binary words?
I know that I need 5 bits to represent a character. All the combinations to encode the 26-letter alphabet will be 2^5? How about the 3 bits that remains from 8 bits?
1
vote
0answers
24 views
Notation for Restriction of Permutation
Suppose $\sigma$ and $\tau$ are permutations such that $\sigma(x)\not=x\implies \sigma(x)=\tau(x)$. Intuitively, I would like to think of $\sigma$ as a restriction (or projection) of $\tau$ onto a ...
0
votes
1answer
13 views
finding out team number
A supervisor has to select a three-member project team from among her 12 employees. Unfortunately, two of the employees cannot work together on the same team. With this restriction, how many different ...
0
votes
1answer
15 views
Probability of a subset occuring within a section of a permutation
I have a list of permutations of a vector of length 24. In each position of the vector is a number between 1 and 24 and repeats are not allowed.
An online tool tells me that this will give ...
0
votes
2answers
34 views
Determine no of combinations for cutting stock algorithm
I have to buy $n$ wooden logs of size 2000 each, from which I have to cut different pieces of smaller size say:
255*10
750*7
550*13
In a manner that cutting will ...
0
votes
1answer
16 views
decomposition of m-cycle in m-1 transpositions
I am searching for a proof. Every m-cycle $\sigma = (x_1 x_2 ... x_m)$ can be expressed as an composition of m-1 transpositions.
I found many formulas, for example:
$\sigma = (x_1 x_2)(x_2 x_3) ... ...
1
vote
1answer
49 views
Are these two permutation matrices equivalent?
Definition: The permutation matrix $P_{ij}$ is the identity matrix with rows $i$ and $j$ reversed. When left-multiplied with another matrix, it exchanges rows $i$ and $j$.
Am I right in thinking that ...
4
votes
3answers
84 views
Distributing identical objects to identical boxes
We have 6 identical things to be distributed in 4 identical boxes such that empty boxes are allowed the find the number of ways to distribute the things ?
4
votes
1answer
65 views
Find a lower bound
Let $M$ be an $N\times N$ symmetric real matrix, and let $J$ be a permutation of the integers from 1 to $N$, with the following properties:
$J:\{1,...,N\}\rightarrow\{1,...,N\}$ is one-to-one.
$J$ ...
3
votes
1answer
47 views
Is there a name for this type of permutation?
Let $J$ be a permutation of the first $N$ integers (1, 2, ..., $N$), so that the permuted sequence reads $(J(1),J(2),...,J(N))$. The function $J$ must of course be a bijection. Additionally, suppose ...
