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

learn more… | top users | synonyms (1)

0
votes
2answers
72 views

Group Theory: How to find all possible images of $ f $?

Let $H$ be a group and suppose that $ f: D_{10} \rightarrow H $ is a homomorphism. How do I describe and justify all the possible images of $f$. $D_{10} = ({1, \sigma, \sigma^2, \sigma^3, \sigma^4, ...
1
vote
0answers
49 views

If $G$ acts such that $|\mbox{fix}(g)|\le 2$ for $g \ne 1$ and $O_p(G) \ne 1$ and $|G_{\alpha}|$ odd. Assertions about Frobenius groups

Let $G$ be a finite group acting nonregularly and transitive on $\Omega$ such that each nontrivial element has at most two fixed points and $|\Omega| \ge 4$. I know three facts: i) If $1 \ne X \le ...
0
votes
0answers
21 views

For a specific subgroup $N$ of index $2$ why $( S \setminus Q ) \cap N = \emptyset$ if $S \in \mbox{Syl}_2(G)$ and $|S : Q| = 2$

Let $G$ be a finite group acting on some set $\Omega$ with $|\Omega|$ even. Let $S \in \mbox{Syl}_2(G)$. Further let $Q \le S$ such that $|S : Q| = 2$ and suppose we have some $x \in S \setminus Q$. ...
1
vote
1answer
15 views

Is it possible to find all subgroup of $Alt_4$ from all subgroup of the permutation group $S_4$?

Is it possible to find all subgroup of $Alt_4$ from all subgroup of the permutation group $S_4$? I think the answer is yes from Lagrange's theorem and Sylow's theorem. Is anyone is able to give ...
1
vote
1answer
54 views

Spivak's curious thoughts about the action of permutations.

Here is an excerpt of Spivak's Differential Geometry. What I do not understand is why he believes $\sigma \cdot (\rho \cdot v) = (\rho\sigma) \cdot v$. Since $\sigma$ and $\rho$ are elements of ...
0
votes
3answers
87 views

Random permutation two sets

We have got two sets $A = \{1, \ldots, k \}$ and $B = \{k, \ldots, 2 k - 1\}$. Let $\sigma$ be random permutation of $\{1, \ldots, n\}$, $n \ge 2k - 1$. Let $A_{max} := max\ \sigma(A)$, $B_{min} := ...
0
votes
1answer
22 views

What segment in 8-bit LED Displays used for Traffic Light timers can be removed causing minimal impact in the readability of the countdown numbers?

I passed by an intersection with traffic lights and noticed that 1 segment of the 8-bit display counter is dimmed (it's not working). When the lowermost segment is dimmed for example, number 4 can ...
0
votes
1answer
19 views

Number of ways to select a president and a secretary with restrictions

Solve using addition principle: A committee composed of Jesse, Bianca, Ray, and Lily is to select a president and a secretary. How many selections are there in which Jesse is president or not an ...
1
vote
0answers
32 views

For a group acting such that each nontrivial element has at most two fixed point, size of orbit of single $2$-power order element

Let $G$ be a nonregular, transitive permutation group on $\Omega$ such that each nontrivial element has at most two fixed points. Suppose $S \in \mbox{Syl}_2(G)$ and that we have $\alpha, \beta \in ...
1
vote
2answers
24 views

Can someone explain what this statement means (Groups and permutations) please?

I'm currently reading notes on a lecture I missed due to not feeling well. These are notes on "Symmetric Groups and Modular Arithmetic Groups". A sentence in the notes says: "For a set $S$, a ...
0
votes
1answer
24 views

Can I place the rubiks3cube pieces in the distorted position I intend to get?

Right now I am trying to get the distorted position like this: in each face only one diagonal is solved and no similar colour is on a face other than the diagonal pieces mentioned previously. For ...
2
votes
1answer
47 views

Prove, by logical reasoning, rather than by formula, the following permutation identities

The formula would have been useful but I am not really good at logical reasoning especially in permutations so I need help from you guys to identify errors in my answer (as well as give hints for part ...
1
vote
0answers
28 views

Describe the centralizer

Exercise from Artin's Algebra. Describe the centralizer $Z (\sigma) $ of the permutation $\sigma = (153)(246)$ in the symmetric group $S_{7} $, and compute the orders of $Z (\sigma) $ and of $C ...
0
votes
0answers
35 views

How to generate the next permutation?

I feel confuse with the next permutation. We have $4$ step for find the next permutation. step $1$: find from right to left and check $a[i] < a[i+1]$. step $2$: find the pivot with condition is ...
7
votes
1answer
60 views

A conceptual problem in group theory

As we all know that in group $S_n$ every pair of distinct disjoint cycles commute .my doubt is is it reverse all true,mean if a pair of distinct cycles commute ,then they have to be disjoint??.i ...
1
vote
0answers
62 views

How to divide $C_n^k$ combinations into $C_n^k\times k/n$ distinct groups if $n/k$ is an integer?

Does there always exist a partition like follows? If $n/k$ is an integer, we can always divide $C_n^k$ combinations into $C_n^k/n*k$ groups such that in each group all of the combinations has no ...
1
vote
1answer
90 views

Confused about Catalan number equation

Why is the Catalan number $$C_{n+1} = C_0 C_n+C_1C_{n-1}+\cdots+C_{n-1}C_1+C_nC_0,$$ and not $$C_{n+1} = C_0C_{n+1} + C_1C_n +\cdots+ C_nC_1 + C_{n+1}C_0 ?$$ In the latter formula, $0+(n+1) = 1+n = ...
0
votes
1answer
43 views

What is coefficient of $x^k$ in $ n! (x/1! + x^2/2! + x^3/3! + … )^n$?

Given $n![\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+...\frac{x^n}{n!} ]^n$, how do I find coefficient of $x^k$ in it ? How to find coefficient in case of above series having infinite terms i.e. $n! ...
1
vote
3answers
33 views

Can someone explain the logic of combinations in this question?

Problem: An ancient human tribe had a hierarchical system where there existed one chief with 2 supporting chiefs, each of whom had 2 equal, inferior officers. If the tribe at one point had 10 ...
1
vote
1answer
77 views

Die roll combinatorics and Yahtzee conundrum

While working out number of ways of getting different outcomes from a $6$ faced die rolled $6$ times, I stumbled upon a simplified approach, illustrated by number of ways to get the patterns below: ...
0
votes
1answer
19 views

Can't get a correct signature of Entringer number

I am stuck with understanding Entringer numbers which: E(n,k) are the number of permutations of {1,2,...,n+1}, starting with k+1, which, after initially falling, alternately fall then rise. ...
0
votes
2answers
43 views

Permutation of 10 numbers, position 1 or position 2 not allowed

We have a permutation of $\pi$ numbers $\{1,2,...,10\}$. Let $A_1$ be $\pi(1)>1$ and $A_2$ be $\pi(2)>2$. (number on position 1 or 2 must be greater than 1 or 2, respectively). What is the ...
0
votes
0answers
24 views

“Healing” a frequency distribution when filling-in a group…

Consider (purely as an example of a frequency distribution) letters in English, which follow a frequency distribution. A 8%, B 1%, C3% and so on. (eg) EF, English Frequency. Say you have to ...
1
vote
1answer
23 views

How to decide what combinatorics method to use?

Problem: Our football team has 10 members, of which only 3 are strong enough to play offensive lineman, while all other positions can be played by anyone. In how many ways can we choose a starting ...
0
votes
1answer
22 views

Intuition for number of ways of distributing $n$ objects in $r$ groups

Suppose I have $n$ objects(non-identical) and I want to distribute them in $r$ groups.Let's say $n=r=10$ .The solution given is $10^{10}$. I cannot understand this result at all. Well I know if I try ...
1
vote
1answer
37 views

Show that a subgroup of $A_5$ that contains a 3 cycle and acts transitively on $1,2,3,4,5$ coincides with $A_5$

I know $A_5$ is simple, so my line of logic would be to prove that a group containing a 3-cycle that acts transitively on 1,2,3,4,5 must be a normal subgroup of $A_5$. I'm just not so sure how to do ...
1
vote
2answers
36 views

Number of non-negative distinct integer solutions of $x+y+z+w=10$

I understand that there are already many questions relating to this, but my question is regarding some concept of mine that should be working but doesn't produce the right result. So, I follow an ...
0
votes
2answers
33 views

Four players choose 7 cards from a 52 card deck with the rest of the cards going back

I'm doing a homework assignment (with answers uploaded from my TA) and have been stumped over this particular answer. There are 4 players. Each player selects 7 cards from the deck. The remaining ...
0
votes
0answers
42 views

Permutations with “explicit grouping”

I am new to combinatorics and I might asking a trivial question: Let $S_a=\left\{1_a, 2_a, 3_a, \dots, n_a\right\}$ and $S_b=\left\{1_b, 2_b, 3_b, \dots, n_b\right\}$. I know that if I want to take ...
1
vote
0answers
29 views

Point stabliziers of primitive permutation groups are maximal primitive subgroups

While reading Kurzweil & Stellmacher's Theory of Finite Groups, chapter 6.6, Primitive subgroups are defined: a group $M$ is primitive if for every If for every non-trivial $N \unlhd M ...
0
votes
3answers
151 views

Number of ways in which $8$ distinct apples can be distributed among $3$ boys?

Number of ways in which $8$ distinguishable apples can be distributed among $3$ boys such that every boy should get at least $1$ apple and at most $4$ apples is $K× _7P_3$ then what is the value of ...
0
votes
1answer
43 views

How many permutations are there of the letters, taken all at a time, if the word

How many permutations are there of the letters, taken all at a time, if the words (a) ASSESSES (b) PATTIVEERANPATTI Ans: (a) $$\frac{8!}{5!\cdot2!} = 168$$ (b) ...
0
votes
0answers
31 views

How many combinations and permutations can we write number as a sum?

I would like to count number of combinations and permutations we can write number $N$ as a sum of positive numbers. For example: Combinations$(3)=3$ ; $(1+1+1, 1+2, 2+1)$ Permutations$(3)=2$ ; ...
0
votes
1answer
103 views

Polygons within a polygon

r-sided polygons are formed by joining the vertices of a n-sided polygon.Find the number of polygons that can be formed,none of whose sides coincide with those of the n-sided polygon? Polygon is ...
0
votes
1answer
22 views

Slightly trickier round robin problem

I have been introduced to this problem recently which to me resembles of round robin, but with an additional rule which might make no solution plausible. The idea is to find combinations of groups ...
1
vote
2answers
54 views

How many numbers with distinct digits are possible product of whose digits is 28?

This is a question asked in India's CAT exam: http://iimcat.blogspot.in/2013/08/number-theory-questions-and-solutions.html How many numbers with distinct digits are possible product of whose ...
0
votes
1answer
26 views

How many ways to arrange 10 values in a vector of length 50?

I have a set of values $1,2,3,4,5,6,7,8,9,10$ that I want to place in vectors of length 50 in all permutations. So far I'm using the permutation formula with $n=50$, $r=10$ as $$ \frac{50!}{(50-10)!} ...
0
votes
2answers
22 views

Permutations of sorted multisets

I am new to combinatorics and might ask a trivial question: Given two ordered sets $S_a=\left\{1_a, 2_a, 3_a, \dots, n_a\right\}$ and $S_b=\left\{1_b, 2_b, 3_b, \dots, n_b\right\}$ How can I ...
1
vote
3answers
55 views

How many possible outcomes are there when 5 similar dices are rolled?

In this question If I consider first total no of outcomes as $6^5$ , then I divided it be ${6\choose5} *5!$ since there are 5 similar dices so the outcome (1 2 3 4 5 ) will be similar to (5 4 3 2 1) ...
0
votes
1answer
26 views

Six digit numbers that are divisible by 3

A question I encountered recently : A six digit number divisible by 3 is to be formed using the digits 0,1,2,3,4 and 5 without repetition. How many number of ways can this be done ? If it asked for ...
0
votes
1answer
69 views

Different ways to arrange a set of numbers, so X can be seen from the left and Y can be seen from the right

Given an set of unique integers of length N. What are number of different ways you can rearrange the array so that, you can only see X numbers of integers from the left and Y numbers of integers from ...
1
vote
0answers
16 views

Eccentricity of vertex in a permutation cycle graph

The permutation cycle graph is defined as follows. Permutation Cycle Graph: A permutation cycle graph for a given permutation $\pi$ of a finite set $V$, is its cycle graph $\Gamma$ such that ...
0
votes
1answer
29 views

Order of permutations

A pack of 2n cards is shuffled by the "interlacing" method, in other words, if the original order is 1, 2, 3, 4,...,2n, the new order after the shuffle is 1, n+1, 2, n+2,... n, 2n. Work out how many ...
11
votes
3answers
433 views

Group of $r$ people at least three people have the same birthday?

What is the probability that in a randomly chosen group of $r$ people at least three people have the same birthday? $\displaystyle 1- \frac{365\cdot364 \cdots(365-r+1)}{365^r}$ $\displaystyle ...
1
vote
3answers
57 views

How many segments can I draw between $17$ points on the circumference of a circle so that each segment intersects all others inside the circle?

A set of $17$ points is chosen on the circumference of a circle. Find the maximum number of segments which can be drawn connecting pairs of points such that each segment intersects all others ...
-1
votes
1answer
28 views

No. Of elements of order 2 in symmetricgroup of degree 5 [closed]

How many elements of order 2 are there in symmetric group $S_5$?
0
votes
1answer
25 views

Action of permutation group on set of numbers is transitive

I would appreciate if someone could please tell their opinion about my proof. I think the proof makes sense, but I don't know if it's rigorous enough. Theorem: Let $S_n$ be a symmetric group of ...
5
votes
4answers
508 views

Minimum Number of Races to win the Formula 1 World Championship

There are a few complications in the allocation of Formula 1 points through the season, such as half points for an incomplete race, and the fact that a team can change drivers mid-way through a ...
1
vote
1answer
52 views

How to choose $n$ $u$-cubes of different colors so that no two of which are in the same level?

Let $n\geq 4$ be an even integer. A cube with edge of length $n$ (briefly, an $n$-cube) is constructed from $n^3$ unit cubes (briefly, $u$-cubes). There are $\frac{n^3}{4}$ different colors given ...
1
vote
1answer
12 views

subgroup generated by a permutation

For instance, if I take the permutation $\alpha=(123)(67)(458) \in S_{10}$, what is the subgroup generated by it? Knowing that the order of $\alpha$ is 6, I already calculated $\alpha ^0, \alpha ^1$ ...