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

learn more… | top users | synonyms

2
votes
1answer
174 views

If $G$ is a transitive permutation group then $\mathrm{fix}(G_\alpha)$ is a block

I am new here and don't know much about Latex so, I attach my question from Permutation Groups by J. Dixon. I hope to get a help for it: 1.6.5 Let $G$ be a transitive subgroup of ...
5
votes
1answer
163 views

Finding what $\langle(135)(246),(12)(34)(56)\rangle\subset S_{6}$ is isomorphic to

I am doing an exercise that asks me to find what $\langle(135)(246),(12)(34)(56)\rangle\subset S_{6}$ is isomorphic to. I am allowed to only use the groups $D_n,S_n,\mathbb{Z}_n$ and the direct sums ...
6
votes
2answers
287 views

Need help in determining where this pascal's triangle-like sequence comes from.

I have a very interesting problem in that a program that I am running has generated a sequence of numbers that act like the pascal's triangle but have somehow built more structure into it. I have been ...
1
vote
2answers
2k views

how many 5-digit numbers satisfy the following conditions

How many five-digit numbers divisible by 11 have the sum of their digits equal to 30? I am able to get the 5-digit numbers divisible by 11 and I am also able to get the five-digit numbers whose sum ...
1
vote
0answers
250 views

Counting question on permutation matrices with rotation and imprinting

Please read question of distinct permutation matrices with rotation at first, then new counting questions are below: For a distinct $N\times N$ zero-symmetry permutation matrix, we could rotate it 3 ...
6
votes
2answers
259 views

Teaching permutations, How to?

I posed this question to my niece while teaching her permutations: Given four balls of different colours, and four place holders to put those balls, in how many ways can you arrange these four ...
3
votes
0answers
190 views

Combinatorics Issue without repetitive combinations

We have 26 Boxes Labeled: Box 1, Box 2, Box 3 and so on. The boxes are in a specific order. We also have 15 rocks. Rocks are all identical. meaning Rock 1 is no different then Rock 2, or does not have ...
2
votes
2answers
106 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 $I$.
3
votes
3answers
271 views

Is there a fast way to determine the conjugacy classes of $S_5$?

I was working towards proving $A_5$ is the only nontrivial normal subgroup of $S_5$. To do this, I wanted to find a set of representatives of conjugacy classes of $S_5$, and their respective orders. ...
1
vote
1answer
210 views

Any permutation in symmetric group n can be rewritten as a composition of transpositions

I just want to show that a permutation can be written as a composition of transpositions. I cannot use cycles.
3
votes
1answer
309 views

How many non-isomorphic permutation selections are on an arbitrary N x N square matrix with rotations applied?

My question is an extension to a classic one: On a square $N \times N$ grid, select exact $N$ cells that satisfy condition: only one cell selected in same row and column. How many solutions will ...
2
votes
1answer
293 views

Number of permutations with a certain number of fixpoints

Given a set of $n$ mutually distinct elements, how many permutations are there such that exactly $k$ of the permuted elements stay at the same place? Example Let's take the set $\{A,B,C,D\}$. The ...
1
vote
1answer
59 views

Why is the holomorph of a group $G$ a group of transformations of $G$?

I'm reading a definition of the holomorph of a group $G$, where it is defined as $G_L\operatorname{Aut}(G)$, where $G_L$ is the group of left translations of $G$, that is, the maps of form ...
1
vote
0answers
150 views

How to find the last non-zero digit in ${^n\!P_k} $?

What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
3
votes
2answers
717 views

What will be total number of solutions of $a+b+c = n$?

Please tell me how to find the total number of intergral solutions of $$ a+b+c=n $$ I already know that total number of solutions will be $(n+3-1)c(3-1)$. But what will be the case when a varies from ...
0
votes
0answers
67 views

$i$ balls to paint $k$ colors, and exactly $k' < k$ colors should be used. How many ways to paint?

Another ball-painting problem: assume that we have $i$ balls (with numbered labels, so order is sensitive), and $k$ different colors. Now we need to paint these balls using these colors so that ...
1
vote
1answer
241 views

The sgn function and permutations

Let $P=\{(i, j)|1\leq i<j\leq n\}. $For $\sigma\in S_n$, define $\operatorname{sgn}\colon S_{n}\rightarrow \{\pm 1\}$ by $$ \operatorname{sgn}(\sigma)=\prod_{(i, j)\in ...
2
votes
2answers
79 views

Can two finite sequences be considered permutations if their products and sums are equal?

Given two finite ordered sequences with possibly non-unique elements all greater than one: $A,B \in \mathcal{Z}_{>1}$. Given that we have: \begin{eqnarray} |A| &= |B| \\ \Pi_{x \in ...
1
vote
1answer
73 views

Cycle types of $S_4$

I understand the possible cycle types of $S_4$ are $(4), (3,1), (2,2), (2,1,1), (1,1,1,1)$, but why are there $6, 8, 3, 6, 1$ of each respectively? Also why do the elements of each cycle type form a ...
0
votes
2answers
170 views

Problems about symmetric groups

We just finished a lesson about determinants and the symmetric group with all what comes with it ( permutations, transpositions etc... ), except we didn't do group theory ( we only see it next year ), ...
0
votes
1answer
177 views

permutation of “counting out”

Josephus problem*: circle=1,2,3,4,5,6,7,8,9,10. count=2. (Beginning at 1) The "last man standing" in this case=9. Order of elimination or permutation (?): 2,4,6,8,10,3,7,1,9 For any size circle and ...
3
votes
1answer
282 views

Understanding fundamental principles of counting.

There are two fundamental principles of counting; Fundamental principle of addition and fundamental principle of multiplication. I often got confused applying them. I know that if there are two ...
2
votes
2answers
164 views

15Puzzle, sum of inversions - what's been summed?

According to this page: http://mathworld.wolfram.com/15Puzzle.html it says that While odd permutations of the puzzle are impossible to solve I've red this article: ...
1
vote
1answer
83 views

What is the number of functions $f : A\rightarrow A, \forall_{x\in{A}} f(f(x))=x$, set $A$ have $n$ distinct elements.

What is the number of functions $f : A\rightarrow A, \forall_{x\in{A}} f(f(x))=x$, set $A$ have $n$ distinct elements. For $A=\{a,b,c,d\}$ we could look on the right-upper triangle of matrix of all ...
1
vote
1answer
254 views

Adjacent transposition - explain in layman terms

I've red this article many-many times, as well as some more supplementary reading on groups, permutations, generating sets etc. and I give up. :( I'm not a mathematician (I study combinatorics, but ...
0
votes
2answers
160 views

Probability of random shuffling of cards

I have a pack of cards and use the following method to shuffle them Pick a random card from the deck and replace the first card with it Put the first card back in the deck Move to the second card ...
2
votes
0answers
31 views

order of elements in a partition using Maple

I determined this whole partition but I just want to have the finer the partition for example: I have this ...
6
votes
1answer
666 views

Conjugacy classes in $A_n$.

Suppose $n$ is a non negative integer $\geq 4$ and $\sigma\in S_n$ a permutation. Conjugacy classes in $S_n$ are completley determined by the cycle structure of $\sigma$. If we let the alternating ...
5
votes
2answers
358 views

permutations of a multiset having symbols with fixed multiplicity

Let $N$ be a multiset of $n$ distinct objects having the same multiplicity $k$. For instance, $N=\{a,\,a,\,b,\,b\}$ where $n=2$ and $k=2$. I was looking for the problem of counting the number of ...
3
votes
1answer
160 views

Find the subgroups of index two of this finite semi-direct product

This older stackoverflow question may be helpful in answering the question that I ask below, although I could not work it out. For $n\geq 1$, let $X=\lbrace 1,2, \ldots ,n \rbrace$, $Y=X \cup (-X)$ ...
2
votes
5answers
248 views

Bijection proof

I got this on an exam and struggled to complete it, could anyone offer a proof? Thanks! Let $X$ be a finite set. Let $f: X \longrightarrow X$ be a bijection. For $n \in \mathbb{Z}^+$, set $$f^n ...
1
vote
2answers
163 views

compositions of permutations

For compositions of permutations on a set $X = \{1,2,3\}$, my lecture notes say that the composition $\phi_2 \phi_1$ is the permutation $\phi_1$ followed by the permutation $\phi_2$. So consider the ...
3
votes
1answer
318 views

Sliding blocks puzzle

Consider a 'game' played on a subset $S$ of an $n^2$ square grid as follows. There are 3 types of pieces, each occupying a square, 1 green, some red and the rest are blue, a move consists of shuffling ...
2
votes
1answer
48 views

A statement about $S_n$: pigeonhole?

Let $n \in \mathbb N$, $n \ge 2$ and let $S_n$ be the symmetric group on $n$ elements. I will call for shortness $I_n := \{1 , \ldots , n\} \subset \mathbb {N}$. Fix $i_0 \in I_n$ and consider the ...
1
vote
0answers
106 views

Random Permutation Poisson proof

Let $F$ be the number of fixed points of a random permutation on $n$ items. Show that as $n$ approaches infinity, the distribution of $F$ approaches a Poisson distribution with a mean $(\lambda)=1$.
0
votes
1answer
486 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 ...
0
votes
1answer
139 views

Permutation & Combination wordproblems

I'm studying Permutation & Combination those days and I've got well understanding the whole chapter but those word-problems related to it can't got them well, not even understand any of them. for ...
5
votes
1answer
3k views

Number of ways of choosing $m$ objects with replacement from $n$ objects

There is a set of $n$ distinct objects. How many possible multisets can we get when choosing $m$ objects with replacement? Note that the elements in a set are unordered and distinct, and the elements ...
1
vote
1answer
152 views

what is the formula to calculate the permutations

I am new to the permutations. I have a problem with me for which I am not able to use proper formula - Problem: There are X boxes in which balls need to be placed. The balls are of two colors - BLUE ...
1
vote
1answer
97 views

Permutation for balls [duplicate]

Possible Duplicate: number of combination in which no two red balls are adjacent. We have $N$ slots. They have to be filled with balls (either green or red), one ball for each slot. ...
2
votes
1answer
333 views

Why does multinomial theorem only works for identical set of objects?

I will elaborate this with an analogy, 15 toys are to be distributed amongst 3 children , such that any child can get any number of toys, so we have to find the number of ways in which we can do so ...
0
votes
3answers
814 views

Number of four digit numbers?

I came across this question, that how many 4 digit numbers $a_1a_2a_3a_4$ are there such that $a_1\geq a_2\geq a_3\geq a_4$ I know the answer for this case, $a_1>a_2>a_3>a_4$ I ...
7
votes
1answer
695 views

number of combination in which no two red balls are adjacent.

given x spaces(you can fit 1 ball in 1 space) and unlimited number of identical red and white balls, find the total number of combinations in which no two red balls are adjacent to each other. i ...
2
votes
2answers
855 views

Irreducibility of the standard representation of $S_n$.

The permutation representation of $S_n$ is $\mathbb C^n$ with elements of $S_n$ permuting the basis vectors $\{e_1, e_2, \ldots, e_n\}$. It has a trivial subrepresentation spanned by the vector $v = ...
1
vote
1answer
785 views

How many combination of numbers

Anyone can give me an idea how I approach calculating the following problem. How many possible valid numbers, where a valid number is any number between 0-9, length of 10 digits, excluding # or *, a ...
2
votes
2answers
676 views

How does one decompose the regular representation of $S_3$?

I need to decompose the regular representation of $S_3$ into irreducible ones. What I know so far is this: $S_3$ is generated by $\tau = (12)$ and $\sigma = (123)$. If $v$ is an eigenvector of ...
2
votes
1answer
1k views

20 books 5 different shelves

So I'm trying to answer this question and am not sure if my answer is correct. In the text book I'm using, this question asked before combinations are even introduced (only permutations) so I'm not ...
-2
votes
1answer
175 views

How many possible ways are there

Suppose i have the given data set of length 11 of scores p=[2, 5, 1 ,2 ,4 ,1 ,6, 5, 2, 2, 1] I want to select 6 ,5 , 5 , 4 , 2 , 2 scores from the data set. How many ways are there? For the above ...
2
votes
1answer
385 views

Simple Dice Rolling Problem

If you play poker dice by simultaneously rolling 5 dice, why is $P\text{{five alike}} =.0008$? I guess I understand the fact that each dice has the probability to land on the same number $1/6$ of the ...
2
votes
2answers
449 views

$n$-permutations with exactly $k$ fixed points

It's easy to deduce the formula for $n$-permutations with exactly $k$ fixed points. The result is similar to $n$-derangement formula and it's equal to $ D_{n,k}= ...