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

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2
votes
2answers
190 views

How do I prove that an order of a cycle is its length?

Let $\sigma$ be a cycle with length $n$ where $\sigma \in S_m$. How do i prove that $|\langle \sigma \rangle |$ is $n$?
0
votes
2answers
108 views

Finding Distinct Elements and Permutation in Partitioned Set

I am having a hard time figuring out where to start on a homework problem. The question is: A set of $nk$ elements is partitioned into $k$ subsets in two ways, each subset having size $n$: one ...
5
votes
3answers
233 views

A Problem of Combinatorics

In how many ways can three distinct numbers be chosen from the set {1,2,3,4....2n} such that the numbers are in increasing arithmetic progression?
0
votes
2answers
51 views

$(34)(123)(456)$ is a cycle. True or False?

I know this is basics, and I understand that $(34)(123)(456)$ is a product of cycles which, I found: $(124563)$. But somehow, I was lost. How do I know if it is indeed a cycle? OR if it isn't? Any ...
0
votes
1answer
24 views

permutations and combination

How many different strings of lights can be created by placing 40 coloured lights on a horizontal string if 12 of them are red, 6 are blue, 14 are green and 8 are yellow? Assume that lights of the same ...
0
votes
2answers
157 views

Bridge hand Combination/Permutation

A Bridge hand consists of 13 cards from a deck of 52 cards. In how many ways can a (bridge) hand consisting of 5 spades(♠), 4 hearts(♥), 4 diamonds(♦) and 0 clubs(♣) be selected?
2
votes
1answer
47 views

In how many ways can letters in mathematics be ordered with restrictions?

I've been stuck on these for a while. Please guide me through all the steps because I actually want to understand this. I've got an exam coming up. Consider the letters in the word "MATHEMATICS". In ...
3
votes
2answers
221 views

Algorithm to compute maximum permutation sum in matrix

Given a matrix $A_{n\times n}$ of real numbers, what fast algorithms do there exist to compute the maximum value of $a_{1,\sigma(1)}+a_{2,\sigma(2)}+\ldots+a_{n,\sigma(n)}$ over all permutations ...
0
votes
1answer
24 views

My proof regarding composition of permutations came to the same conclusion as the answer sheet, but through different methods. Is it valid?

Let $S_3$ be a set of all permutations of elements in $\{1,2,3\}$. Prove that there doesn't exist f $\in S_3$ where $\{f,f^2,f^3,f^4,f^5,f^6\} = S_3$. Where $f^n = f \circ f \circ \:... \circ \:f$ ...
2
votes
2answers
149 views

The Cayley Representation Theorem.

This theorem states that "Any group is isomorphic to a subgroup of a group permutations." I only ask if someone could provide a simple example so that i can fully understand this theorem.
0
votes
1answer
83 views

Do disjoint cycles commute?

When a given set is finite it is clear. I'm asking the general case. Let $X$ is an arbitrary set. Let $\sigma,\tau$ be disjoint cycles on $X$. Then do they commute?
0
votes
1answer
62 views

How many way can 7 friends line up if there are certain conditions?

How many ways can 7 friends line up if Ann, Beth, and Chris have to stand next to each other where Ann is ahead of Beth and Beth is ahead of Chris? Would it simply be $5*4*3*2*1=120$ ways? Expanding ...
0
votes
1answer
62 views

What is the definition of “disjoint cycles”?

I'm the one who thinks clear definition(clear with meta-language) is very important for doing mathematics. Below, i list my definitions for cycle and orbit. Let $X$ be a nonempty set. Let ...
0
votes
2answers
71 views

What is “the orbit of a permutation”? Is the term “orbit” consistent with that for group action?

reference: What is the orbit of a permutation? To be honest, i don't understand the answer in the link. The orbit of a group action is defined as follows: Let $G$ be a group acting on a set $X$. ...
0
votes
1answer
39 views

Prove that $sgn(\sigma_1 \circ \sigma_2) = sgn(\sigma_1)sgn(\sigma_2)$

Lete $n\in \mathbb{N}$. Show that the transformation $$sgn: S_n \rightarrow \{\pm 1\}$$ (where $S_n$ is the set of all permutations of the integers in the set $\{1,...,n\}$),given by $\sigma \mapsto ...
0
votes
1answer
75 views

Analog of Birkhoff's theorem for doubly stochastic matrices

Birkhoff's theorem states that extreme point of the set of doubly stochastic matrices are permutation matrices. An $n \times n$ matrix $A$ is doubly stochastic if each row and column sums to 1. What ...
-2
votes
1answer
69 views

conjugacy classes and order of group

Suppost that $k_G(A)$ denotes the number of conjugacy classes of $G$ that intersects $A$ non-trivially ($A$ is an arbitrary subset of $G$) and $M=G^{'}Z(G)$. Also suppose that $G$ is non-solvable, ...
0
votes
2answers
47 views
1
vote
1answer
107 views

Permutations of two photo frames

Please help with this permutations question. I'm trying to use the permutation formula to calculate it but don't know where to begin: $$\frac{n!}{(n-r)!}$$ Here's the problem: ...
1
vote
1answer
90 views

Envelopes and Mailboxes

We suppose $n$ and $p$ are two positive integers. A) In how many ways can you divide $p$ identical envelopes in $n$ mailboxes? (Each mailbox can hold several envelopes at the same time) B) In how ...
0
votes
1answer
30 views

Count no. Of ways

If $n$ identical balls put into $m$ identical boxes, how many ways it can be done, provided that boxes may be empty and all balls have to be put into these boxes at each time.
0
votes
1answer
45 views

Probability of item distribution with a restriction

I'm having a hard time analyzing my research data, and was wondering if anyone had any suggestions? I've reworded the question so it is presented more like a statistics problem. There are $x$ number ...
1
vote
2answers
49 views

Sum of Binomial Series of form $\binom{2000}{3k-1}$

Find the Value of $$ \binom{2000}{2}+\binom{2000}{5}+\binom{2000}{8}+\cdots+\binom{2000}{1997}+\binom{2000}{2000}$$
2
votes
2answers
1k views

(12345) is an even permutation of S_5. True or False?

The answer i had for this question was True, yet i'm not sure. Well, from what I know so far was that: $(12345)$ can be expressed as a number of 4 transpositions such as: $(12)(23)(34)(45)$ which is ...
0
votes
1answer
46 views

Conjugate subgroups of $S_4$

$A = \langle (1,2,3),(1,2)\rangle$ $B = \langle (1,2,4),(1,2)\rangle$ $C = \langle (1,3,4),(1,3)\rangle$ $D = \langle (2,3,4),(2,3)\rangle$ I want to proof that these subgroups of $S_4$ ( which ...
2
votes
3answers
2k views

How do I solve for n in this permutation question?

I have the following question: Solve for n: $$_nP_3 = 6_{n-1}P_2$$ I don't know how I should begin to tackle this problem? Any tips/help would be appreciated.
2
votes
2answers
58 views

Proof that $\det(A)=\det(A^T)$ using permutations.

I'm reading a proof for the identity $\det(A) = \det(A^T)$ and I'm trying to udnerstand why the following rows are equivalent: $$\eqalign{ & \det ({A^T}) = \sum\limits_{\pi \in {S_n}} ...
0
votes
0answers
36 views

Fixed points and permutations.

Let $\psi ,\varphi \in {S_n}$ two permutations. Let $M$ a matrix such that $a_{i,j}=1$ iff $i=\sigma(j)$ where $\sigma \in S_n$ ($0$, otherwise) I already showed that $tr(M) = \left| {\left\{ {k \in ...
3
votes
0answers
32 views

(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
0
votes
2answers
869 views

Number of ways to sit 6 girls and 6 boys together with no two girls together.

As the title of the question explains: What I thought on the very first instant was that we will make them sit alternate hence the answer will be 2 * 6! * 6! But ...
0
votes
1answer
299 views

four digit numbers that have at least one of their digits repeated [closed]

The number of four digit telephone numbers that have at least one of their digits repeated is 9000 4464 4000 3986
2
votes
1answer
87 views

Finding the maximum possible order for an element in $S_5$

I understand that you have to write out all the disjoint cycles and then take the least common multiple which yields the highest order. But my question is, do I have to write all elements of $S_5$, ...
1
vote
1answer
71 views

How many distinct elements does a group of permutation on 3 letters have?

I am having some problems solving a problem similar to this. So i tried making it a more simpler problem. I really don't know how to approach this kind of problem. A hint would be very much ...
4
votes
1answer
72 views

let $S_4$ denote the group of permutations of $\{1,2,3,4\}$ and let $H$ be a sub group of $S_4$ of order $6$ .

Let $S_4$ denote the group of permutations of $\{1,2,3,4\}$ and let $H$ be a sub group of $S_4$ of order $6$ . Show that $\exists~ i \in \{1,2,3,4\}$ which is fixed by each element of $H$. Attempt: ...
0
votes
3answers
328 views

Forming 4-digit odd numbers under certain rules [closed]

How many four-digit odd numbers can be formed such that every $"3"$ in the number is followed by a $"6"$? A) 108 B) 2592 C) 2696 D) 2700
0
votes
2answers
151 views

Coloring vertices of a square

Using four colors, red, white, blue and green, in how many ways can the vertices of a square be colored? Assume that reflections and rotations are allowed, meaning that if you examine a square from ...
1
vote
1answer
48 views

How to compute cyclic notation (23)(1)

I seem to become confused whence computing simple cyclic notations as such. From my understanding, the rule goes by starting from the right and to the left. However by doing this I only end up with ...
0
votes
1answer
223 views

Number of seating arrangements in 5 cars

An exercise from Introductory Combinatorics by Richard A.Brualdi: A roller coaster has five cars, each containing four seats, two in front and two in back. There are 20 people ready for a ride. ...
0
votes
1answer
17 views

number of elements in unsortet case

I have a group M with Mn different elements. How many unique combinations can I make out of this in an n digit system when order is no importance. For example if M = {1 2} & n = 3 ...
1
vote
1answer
290 views

How many ways are there to place these books on the shelves?

You are given 5 books and 7 bookshelves. How many ways are there to place these books on the shelves? (The order on the shelves matters.) I want to say $7^5$ since there are 7 possible shelves and ...
1
vote
1answer
42 views

common homomorphic image and direct products

Let $G \le S^\Omega$, and let $\Delta \dot{\cup} \Gamma$ be a nontrivial partition of $\Omega$. Suppose $\Delta$ (and hence $\Gamma$) is $G$-invariant. Let $G^\Delta$ denote the restriction of $G$ ...
1
vote
1answer
39 views

triangles and lines

There are 12 points in a plane. If 4 of them are on a straight line and no other 3 points are on a straight line, then find the difference between the number of triangles and the number of straight ...
0
votes
1answer
40 views

arrangement of balls in bowls

There are five bowls numbered $1$ to $5$. There are $5$ green balls and $6$ black balls. Each bowl is to be filled by either a green or black ball and no two adjacent bowls can be filled by green ...
1
vote
3answers
97 views

Writing a permutation group in 2 row notation

I have a permutation group in $S_7$, namely: $$(12345)(137)(56)$$ How do I write this in two row notation? I am to write it as disjoint cycles and then as transpositions but I feel better working in ...
0
votes
1answer
23 views

Need a counterexample: A product of a cycle with itself will in general be a cycle.

Need a counterexample: A product of a cycle with itself will in general be a cycle. I was thinking something in S_4.
1
vote
1answer
45 views

how can you count number of digits used in numbers from -2^127 to (2^(127) - 1)

There are numbers from -2^127 to (2^127)-1. I want to count the number of digits used in all the numbers. For example If I have numbers from -100 to 100 then number of digits used is $2*(1*9 + ...
2
votes
2answers
54 views

Kernel of $\phi:G \rightarrow \operatorname{Sym}(S)$ Group actions

$\operatorname{Sym}(S) == \text{All permutations of the set }S$. Prove $\ker(\phi)=\bigcap_{x\in S}G_x$ where $G_x$ is the stabilizer of $x$. Let $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
9
votes
1answer
30k views

How many possible combinations in 8 character password?

I need to calculate the possible combinations for 8 characters password. The password must contain at least one of the following: (lower case letters, upper case letters, digits, punctuations, special ...
1
vote
1answer
84 views

Number of permutations of AABBBCC, taking 7 letters at a time, when repititions are allowed

What is the number of permutations of the word AABBBCC, taking 7 letters at a time, repetitions being allowed? I think it should be $3^7$, but I can't see why. Also what would be the number of ...
2
votes
0answers
35 views

How many commutative block ciphers are there?

Let $K$ and $M$ and be two finite sets. Let $(G,\circ)$ be the group of permutations over $M$ under composition. Let a (implicitly: block) cipher with key in $K$ and message in $M$ be any application ...