Tagged Questions
1
vote
1answer
30 views
Computing the order, inverse, and parity of a permutation
How do you compute the order, inverse and parity of $\alpha=(12)(43)(13542)(15)(13)(23)$? Please explain all steps taken to get the answer.
I guess my thought process was to first put it into a ...
1
vote
2answers
66 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 ...
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
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
2answers
54 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 ...
1
vote
1answer
50 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 ...
1
vote
5answers
77 views
Prove that sgn$(\sigma \tau) = $sgn$(\sigma)$sgn$(\tau)$
Prove that sign$(\sigma \tau)$ = sign$(\sigma)$sign$(\tau)$ for any permutations $\sigma, \tau \in S_n$.
I think the two thing's I'm trying to show are:
If sign$(\sigma)$ = sign$(\tau) = \pm 1 ...
1
vote
2answers
43 views
Inclusion & Exclusion: In how many permutations of the digits $0,…,9$ there's no continuity of 7 digits or more?
In how many permutations of the digits $0,...,9$ there's no continuity of 7 digits or more?
(Ex. the number 203456789 1 should not be counted)
I believe that the basic case, for the inclusion ...
1
vote
1answer
80 views
Exponential generating function for permutations with descent set whose least element is even
Let $E(n)$ be the number of permutations $w\in S_n$ such that the least element of the set $Des(w)\cup \{n\}$ is even, where $Des(w)$ is the descent set of $w$. I need to find the exponential ...
1
vote
3answers
63 views
Restricted Permutations
I have a problem in which 4 music books, 5 education books and 2 medicine books need to be arranged on a shelf. In how many ways can this be done if only the music books must be kept together and all ...
0
votes
1answer
47 views
Number of $k$-cycles in permutations of $[2k]$?
What is the expectation of the number of $k$-cycles in a randomly selected permutation of $[2k] = {1,2, . . . ,2k}$?
0
votes
2answers
147 views
A step in finding the determinant of transpose of a matrix
The following question involves the permutation group, which I am horrible at handling. Any help will be greatly appreciated.
Let $A$ be an $n \times n$ matrix with entries $(a_{ij})_{i = 1,2 \cdots, ...
2
votes
1answer
80 views
Questions about products of $p$-cycles.
Let $p$ be a prime and let $n$ be an integer such that $n \le p$.
a) Let $\alpha$ be a p-cycle in $S_n$. Prove that $\alpha^j$ is a p-cycle for $1 \le j \le p − 1$.
b) Assume that $2p \le ...
0
votes
3answers
99 views
Examples of Permutation
$(1)$ How many elements of order $5$ are in $S_7$(symmetric group)?
$(2)$ How many odd permutation of order $4$ does $S_4$ have?
$(3)$ If $\beta \in S_7$ and $\beta^4=(2143567)$ then find $\beta$
...
2
votes
3answers
187 views
Let $\alpha=(1234)(5876)$ and $\beta=(1537)(2648)$ belong to $S_8$.
Let $\alpha=(1234)(5876)$ and $\beta=(1537)(2648)$ belong to $S_8$. Determine whether there exists a subgroup of $S_8$ that contains $\alpha$ and $\beta$ and is isomorphic to $D_4$.
1
vote
5answers
175 views
Expression as a product of disjoint cycles
Let $\alpha = (9312)(496)(37215) \in S_n, n \ge 9$. Express $\alpha$ as a product of disjoint cycles.
I know this is probably a really easy question, but my professor didn't elaborate on how to ...
3
votes
2answers
104 views
Suppose that $\alpha$ and $\beta$ in $S_4$, $\alpha (3) = 4$, $\alpha\beta = (3412)$, $\beta\alpha = (3241)$, find $\alpha$ and $\beta$
Suppose that $\alpha,\beta \in S_4, \,\alpha (3) = 4,\, \alpha\beta = (3412), \,\beta\alpha = (3241).\;$ Find $\alpha\,$ and $\, \beta$.
My attempt:
Ok so I know
$\alpha(3) = 4$ and ...
0
votes
2answers
78 views
A probability problem - probability of one card being red and the other one being black.
Consider a deck of 50 playing cards (2 cards missing). What is the probability that one of them is red and the other one is black? I've got two solutions which one is correct ?
Let $R$ represent red ...
1
vote
2answers
293 views
Combinatorics and Probability Problem
The problem I am working on is:
An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession.
a.How many different possible PINs are there if ...
0
votes
1answer
37 views
Probability of A Specific Type of Experiment Occuring
The problem is:
An experimenter is studying the effects of temperature, pres-sure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different ...
-1
votes
3answers
71 views
$S_4$ is not isomorphic to $D_{12}$
Why $S_4$ is not isomorphic to $D_{12}$? where $S_4$ is symmetric group and $D_{12}$ is dihedral group of order $12$.here number of element are same $S_4$ has $4!=24$ elements and $D_{12}$ has also ...
2
votes
1answer
253 views
How many ascending and descending numbers are between 1000 and 9999?
I'm working on some homework right now, and I've gotten stumped. Here's the question I'm on:
How many of the 9000 four-digit integers 1000, 1001, 1002, . . . , 9998, 9999 have four distinct digits ...
1
vote
2answers
55 views
Given $\tau\in{S_n} $ a cycle of length $m$ that $\tau^{m} = e$
I'm trying to rigorously prove that given a cycle $\tau\in{S_n} $ of length $m$, then $\tau^{m} = e$ where $e$ is identity.
The funny thing is that I know why it works and understand it intuitively ...
-2
votes
4answers
207 views
How many numbers lying between 10 and 100 can be formed with the help of digits 1,3,5,7,9?
Please help me!! i don't know how resolve this problem.
This is exercise from permutations in my school. hope you guys can help me :)
1
vote
2answers
48 views
combinations, how many ways are there?
how many ways are there to put 36 non-distinguishable balls in 15 distinguishable buckets? This is what I thought: suppose the balls are distinguishable. every time you want to put a ball in a bucket, ...
1
vote
1answer
107 views
3-letter arrangement for “silly” - Permutations homework
We a homework question that asks us to find the 3 letter arrangement of the word "Silly".
Here is the exact question.
How many three-letter arrangements are there of the letters taken
from the ...
0
votes
1answer
175 views
Ten people are seated at a rectangular table - Permutations homework
I got the following question for homework.
Ten people are to be seated at a rectangular
table for dinner. Tanya will sit at the head of
the table. Henry must not sit beside either
Wilson or ...
0
votes
1answer
222 views
how many 2x2 matrices are invertible in mod p
I am trying to solve this problem for homework but unable to get anything. The question is to find the number of invertible 2x2 matrices in mod p? Each entery can bee from the set ...
1
vote
1answer
117 views
Combination , finding number of cases
How many strings with five or more characters can be formed from the letters SEERESS?
Correct me Please,
This is my answer for $6$ characters :
$\frac{6!}{3!\cdot3!} = 20$
$\frac{6!}{3!\cdot2!} = ...
0
votes
1answer
101 views
Permutations & Functions
This is an assignment question I received a week ago.
A function $f:\{1, 2, \dots ,n\} \to \{1, 2, \dots, n\}$ which is a bijection is also called a permutation. Let $P_n$ be the set of all ...
1
vote
1answer
136 views
How many strings of length 12 can we compose using letters A, B, C, and D if every letter should appear at least once?
How many strings of length 12 can we compose using letters A,B,C, and D if every letter
should appear at least once?
can someone walk me through this? I believe using the concept of the sieve formula ...
0
votes
1answer
35 views
Counting coin tosses
Here's a question I got as a homework assignment:
A coin is tossed $k$ times. At some point the result was $T$, and then
later on it was $H$. What are the number of possible permutations?
So, ...
2
votes
2answers
53 views
Permutation without a subword repeat
I am asked the following:
Let m, n, and r be non-negative integers.
How many distinct "words" are there consisting of m occurrences of
the letter A, n occurrences of the letter B, r ...
1
vote
1answer
51 views
Question about cycle decompositions
The Question: Prove that (1 2) cannot be written as the product of 3 disjoint cycles.
The Attempt: Suppose (1 2) has a cycle decomposition into 3 disjoint cycles $m_1, m_2$, and $m_3$. Then (1 2) = ...
1
vote
1answer
79 views
Help needed in proving a theorem on a permutation that is the product of dis. cycles of prime length.
So we were given the following proof to do:
Let $ p $ and $ q $ be distinct primes. Suppose $ \alpha $ is a permutation of $ S_n $ and suppose $ \alpha = \gamma_1 \gamma_2 $ where $ \gamma_1 $ and $ ...
0
votes
0answers
244 views
Find a Lipschitz constant
Please help me to find a Lipschitz constant.
Let $S_n$ be a group of permutations of the set $\{1, \ldots, n\}$. Let $a=(a_1, \ldots, a_{2M})$ be a real valued vector with $n$ non-zeroes entries, $M ...
1
vote
2answers
67 views
Number of Sets problem
A class is attended by $n$ sophomores, $n$ juniors, and $n$ seniors. In how many
ways can these students form $n$ sets of three people each if each set is to contain a
sophomore, a junior, and a ...
2
votes
2answers
137 views
Directed Graphs on Relations - Set Theory
These questions were from an assignment I had some time ago but the solutions were not provided.
A permutation $P$ on a finite set $A$ is a binary relation with the property that for each $a∈A$, ...
1
vote
2answers
102 views
Number of ways of selecting 4 numbers from 20 numbers under certain condition
Out of 20 consecutive natural numbers, in how many ways 4 numbers can be selected such that any two selected number differ by at least 3.
I am not able to proceed further than writing domain of each ...
4
votes
2answers
206 views
How many 4 worded sentences can a list of 5 words make if two of them must be in that sentence?
Suppose we have:
I am new at this - (5 words)
how many 4 worded sentences can we make with this if "new" and "this" must appear in the sentence.
I think its :
.# of sentences we can make with any ...
2
votes
0answers
228 views
sum with permutations
Let $a$ be vector in $R^{2m}$. And let $S_{2m}$ be group of all permutations on the set $\{1,\dots,2m\}$.
I would like to calculate
$$
\sup_{\pi\in S_{2m}}\sum_{d(\sigma, ...
4
votes
1answer
264 views
question involving Markov chain
Let $S_{2m}$ be the group of all permutations $\pi$ of $\{1, 2, \ldots, 2m\}$. The following transition kernel $S$ generates the random transposition walk
$$
Ch(\pi, \pi')= \begin{cases}
\frac{1}{2m} ...
2
votes
2answers
89 views
If $G\subseteq S_n$ is a subgroup acting transitively on $\{1,\ldots,n\}$, then a nontrivial normal subgroup $N\subseteq G$ has no fixed points
Let $G$ be a subgroup of $S_n$, which acts transitively on $I= \{1, \ldots, n \}$. Let $N$ be a nontrivial normal subgroup of $G$. Then $N$ has no fixed points in $I$.
0
votes
1answer
232 views
Random permutation problem
Let $\pi$ be a random permutation of $n$ objects and let $ T := \text{the number of transpositions in } \pi $. Use Chebychev's Inequality to find an upper bound for $T\geqslant k$.
Okay the problem ...
1
vote
1answer
208 views
Combinatorics, arrangements (edited)
"How many ways can the letters in the word SLUMGULLION be arranged so that the three L’s precede all the other consonants?"
My work is below: Can someone also solve this ONLY using the multiplication ...
0
votes
1answer
180 views
What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?
What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?
I believe the answer for this is 6. As we can write the group elements as below
(a)(b)(C)
(ab)(c)
...
0
votes
3answers
81 views
Number of ways in which $38808$ can be expressed as a product of 2 coprime factors?
Number of ways in which $38808$ can be expressed as a product of $2$ coprime factors ?
the answer given is $8$ ways,
what I did was,
$$38808 = 2^3 \times 3^2 \times 7^2 \times 11$$
so the number ...
3
votes
2answers
174 views
Confused about permutation cycles
For some reason I'm finding permutation cycles to be strange and hard to deal with.
Let $\alpha$ and $\beta$ be cycles of odd length (not disjoint). Prove that if $\alpha^2 = \beta^2$, then ...
1
vote
1answer
573 views
Probability that no two people get off elevator on same floor
Seven people enter the elevator on the first floor of a $12$ story building. What is the probability that no two will get off on the same floor? You may assume that all floors are equally likely and ...
2
votes
1answer
73 views
Given an permutation $a$ in $S_8$ how to find $a^{14}$?
Given the following exchange in $S_8$:
$$a = \left(\begin{array}{cccccccc}
1& 2& 3& 4& 5& 6& 7& 8\\
2& 5& 8& 3& 1& 7& 6& 4
...
