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

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3answers
71 views

How do i prove how $S_5$ is generated by a two cycle and a five cycle?

How do I prove that $S_5$ (the permutation group on five letters) can be generated by a two-cycle $(12)$ and a five cycle $(12345)$?
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0answers
20 views

Count of ascending triplets

I would like to solve a problem where in I am given a set of numbers and I have to find the count of all ascending triplets which can be derived from that set. For eg: ...
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1answer
71 views

Permutation + Combination

this is a question I had on my midterm, and I can't seem to be sure what the answer is and our professor did not post the solutions, therefore I cannot make sure I got it right (or wrong). How many ...
2
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1answer
61 views

Possible Ways to reach a Sum

Imagine that I have a N long set of numbers. I would like to know the possible ways that I could reach a specific sum using only the numbers in my set. As an example: ...
2
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0answers
28 views

Permutation elections

During a local campaign eight republician and five democratic candidates are nominated for president. a) If president to be one of thias candidates, how many possibilities are there for the eventual ...
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1answer
57 views

Group Theory, Permutations.

Let $x := (175)(2436)$ and $y := (1234567)$ Compute the elements $x^{-1}yx$ and $y^{-1}xy$? And if $z := (1326745)$ for which $u$ is there an element $v$ such that $v^{-1}zv=u$. If $u$ exists find ...
6
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1answer
233 views

Special Products of Transpositions

[Edit. Significantly expanded to add examples and (I hope) clarification. Feel free to skim by reading the gray boxes.] A colleague asked me for insights on a collection of special permutations, ...
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4answers
55 views

Evaluate the value of $n$ in $\, ^{n+2} P_5=18 \times \, ^{n}P_4$

Suppose $\, ^{n+2} P_5=18 \times \, ^{n}P_4$. How can we evaluate n? I tried expanding it like this: $$(n+2)(n+1)(n+0)(n-1)(n-2)=18 \cdot n(n-1)(n-2)(n-3)$$ Then I try multiplying all of them ...
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2answers
26 views

Find out in how many ways the operation can be performed?

i) In how many ways can a committee of $5$ or more be formed from 12 persons? ii) In how many ways can a committee of $5$ be formed from 12 persons if only two of a group of $3$ persons must always ...
0
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1answer
41 views

Is commutation transitive for permutations?

Let $f,g,h$ be bijections from some set $X$ to itself, i.e. they are permutations of $X$. We say that $f$ and $g$ commute if $f\circ g=g\circ f$. Is it the case, in general, that if $f$ commutes with ...
0
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1answer
51 views

How many ways can we place two types of balls on a circle?

There are $a$ red balls and $b$ blue balls, and I have to place all of these balls on circumference of a circle. The balls with the same color are indistinguishable. I thought the answer would be ...
5
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2answers
92 views

probability of sequence of integers

Suppose you have numbers from 1 to 10. You can choose four of them but the contiguous numbers should have an absolute difference greater than 1. For example, you can choose $$ 1-3-6-10$$ $$ 4-2-5-1$$ ...
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0answers
71 views

Number of permutations with a given constraint

Let $\Pi$ be the set of all permutations of the set $\left\{1 \ldots n\right\}$. Of course I know the cardinal of $\Pi$ is $n!$. I am trying to compute the number of permutations $\pi = \left\{ ...
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0answers
112 views

Proof “correctness” : Cycle structure of conjugate permutations

My Algebra lecturer is a very strict about proofs(w.r.t Completeness , correctness and format ) more so than I have encountered in the past or any of my lecturers of the courses I am take concurrent. ...
0
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1answer
61 views

How many ways of selecting from identical pairs?

My question is with regards to combinations and permutations. How many ways are there to select n unique objects from x number of identical object pairs? To make this question clearer, here is a ...
7
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3answers
186 views

Is there a “natural” / “categorical” definition of the “parity” of a permutation?

Given a permutation $\sigma$ on $n$ elements (i.e. $\sigma \in S_n$), there is a notion of "parity" (or "sign" or "signature") of $\sigma$, which can be defined in several equivalent ways (look here). ...
0
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1answer
153 views

Finding the 'n'th k-permutation of a set, and finding 'n' for a given k-permutation (lexicographical ordering)

Suppose you have a set, and want to order the k-permutations of the set (for example, the permutations of 5 elements of the set {1, 2, 3, ..., 16}). Is there a fast way of finding 'n' (the ...
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1answer
135 views

Permutations expressed as product of transpositions

There is a theorem that states that all permutations can be expressed as a product of transpositions. I have a couple of questions about this theorem: Does the product which is equal to the ...
1
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1answer
133 views

Permutation group of $2^n$ binary numbers

Let $R=\{0,1\}$ and let $D$ be the set of the $2^n$ binary numbers consisting of $n$ bits. Now we apply a permutation $\pi_{i_1i_2...i_n}$ in $D$ to each element $i_1i_2...i_n$ in $D$, which is ...
1
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1answer
116 views

Hash functions for unordered data

I am interested in stream hash functions for unordered data: Hash functions $g$ on lists which, given a two lists of integers $L_1, L_2$ where $L_2$ is a permutation of $L_1$, $g(L_1)=g(L_2)$. I can ...
0
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1answer
35 views

AM I doing this right? - How many binary words of length 8 are there that contain at least six 1's?

How many binary words of length 8 are there that contain at least six 1's? This is what I have: 8!/6!2! = 28 words Is this the correct answer?
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1answer
50 views

How to mathematically calculate the indistinguisable and distinct of the following permutation problems?

I'm having trouble calculating how many indistinguishable and distinct solutions there are for each problems. I'm pretty confident with some of my solutions, but could anyone show me mathematically ...
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2answers
65 views

Does the alternating group $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$?

What are all the possible orders of elements in the group $A_5$? Does $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$? How about $\Bbb Z_{10}$? How about $\Bbb Z_5$? Justify your answers. I've ...
2
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2answers
197 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 $n$?
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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
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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?
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2answers
53 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
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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
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2answers
163 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
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1answer
48 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
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2answers
234 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
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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
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2answers
155 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
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1answer
87 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
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1answer
64 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
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1answer
64 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
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2answers
73 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
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1answer
40 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
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1answer
78 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 ...
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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
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2answers
48 views
1
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1answer
119 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
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1answer
95 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
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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
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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
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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
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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
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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
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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}} ...