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

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Probability of higher occurrence of an element within a random permutation with repetition

I generate $n$ random numbers, each one from a set $X={1,\ldots,N}$, where $n\geq N$. This results in a random permutation with repetition of length $n$ over $X$. Ideally for me, each number of $X$ ...
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1answer
53 views

Number of rules in my fuzzy logic

I have 6 variables with 4 membership functions such as "tiny,small,large,huge". I tried to write the rules and came up with 200 rules but the combinations are killing me and it is still incomplete. ...
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2answers
174 views

What is the smallest alphanumeric string that has 10 million permutations?

I'm aiming to create UUIDs, for a project I'm working on. The standard UUID generators create a very long strings. I'm only anticipating a maximum of 10 million uses and because I'm storing that many ...
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3answers
803 views

Number of ways to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball

How many ways are there to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball? My approach is $ \binom{5}{3} 3! $ + $ \binom{2}{2} \binom{3}{2}2!$ ...
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5answers
123 views

Order of Permutation : If $\tau \in S_n$ has order $m$, then $\sigma \tau\sigma^{-1}$ has also order $m$.

I dont understand the following very simple statement: If $\tau \in S_n$ has order $m$, then $\sigma \tau\sigma^{-1}$ has also order $m$. The proof is: Suppose $\tau$ has order $m$. $(\sigma \tau ...
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3answers
100 views

Need help with combinatorics question(probably cyclical permutation)

A human invites 6 of his friends to a meeting. In how many different arrangements they along with the human's wife can sit at a round table if the hosts and the wife always sit together? Is this a ...
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63 views

functions representable as a sum of two permutations

I am trying to prove that for every function $f:\mathbb{Z}/n\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ satisfying $\sum_if(i)=0$, there exist permutations $\pi_1, \pi_2:\mathbb{Z}/n\mathbb{Z}\to ...
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1answer
75 views

Arranging marbles in a row

Samson has $5$ identical blue marbles, $11$ identical white marbles and $4$ identical red marbles which he wants to arrange randomly in a row. What is the probability that: every red marble will ...
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1answer
84 views

linear arrangements in a row

there are 7 teams A,B,C,D,E,F, each with 5 members. In how many ways can the 35 people be made to sit in a row such that every F team member sits next to i) at least one team G member 2^5*5!*30! (ANS ...
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0answers
124 views

Is there a name for such matrices

Let $Z$ be a $K \times K$ matrix. All its left diagonal elements are zero. Further it satisfies the following properties: 1.) $Z[ i ][ j ] = Z[j][i] > 0$ for $1 ≤ i < j ≤ K$. 2.) $Z[i][j]$ ...
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2answers
174 views

the product of a matrix and a permutation matrix

Can a permutation matrix ($P$) be used to change the rank of another matrix ($M$)? Is there any literature to this effect, or to the contrary? I've tried a few small examples and the resulting matrix ...
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2answers
975 views

How to determine the parity of a permutation by its cycle decomposition

If one is given the length of a permutation and the number of cycles, is it possible to determine the parity of the permutation? Oddly enough, there's no definition in the text I'm reading for ...
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5answers
220 views

finding overlapping permutations

I have a data set $3\; 4\; 5\; 6\; 7\; 8\; 9$ I want to find all the permutations that can be formed using this such that neither $7$ nor $8$ is adjacent to $9$.
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1answer
74 views

The symmetric group $S_4$ and cycles as product of permutations

i have a question, i think it is very easy but i didn't see it: I know that: $$S_4=\langle(1~~2),(2~~3),(3~~4)\rangle$$ Now i want to write $(1~~2~~3)$ and $(1~~2~~3~~4)$ as products of them. ...
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4answers
649 views

distributing z different objects among k people almost evenly

We have z objects (all different), and we want to distribute them among k people ( k < = z ) so that the distribution is ...
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3answers
71 views

probablity of selecting balls

If i have three pots with balls in them as follow. 1st: 2 red 4 black 2nd: 2 red 12 black 3rd: 2 red 4 black what is the chance of getting exactly 2 black ...
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0answers
173 views

balls and bins problem: how to calculate the number of arrangements corresponding to a specific ball sequence before all bins are overflowed?

Considering throwing balls into $k$ bins where each bin can hold $m$ balls at most. Every time a ball is thrown into a bin and we get "$0$" if the bin is not overflowed; Otherwise, we get "$1$". This ...
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1answer
139 views

How many selection ways are possible?

There are 6 girls and 5 boys. You need to make a team of 4 persons. In how many ways can you form this team? One way is to select 0,1,2,3,4 girls in team, but this is a very lengthy process. Is there ...
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2answers
148 views

The probability of data loss

There are 300 files, each of which has 3 copies. Evenly and (by some mechanism)randomly distribute all the 900 files into 10 hard drives such that no drive will contain both a file and its copy. Now 3 ...
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2answers
149 views

Is there a formula in permutations and combinations if we are to find the sum of number of 1's in binary expansion of a number from 1 to n

We are given $N$. Suppose $f(x) =$ number of $1$'s in the binary expansion of $x$. We have to calculate $f(1) +f(2) +f(3)+ \dots +f(N)$. So is there a formula for this sum directly in terms of ...
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168 views

Confused about the group of permutations $S_{n}$

In an exercise, I must prove $S_{n}$ is generated by 2 elements. I'll ignore here the trivial case $n = 1$. Let $I_{n} = \{1, 2, 3, ..., n\}$. I then defined $f : I_{n} \rightarrow I_{n}$ by $f(1) = ...
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1answer
118 views

Element of order $2n$ in symmetric group $S_n$

I've been recently reading some articles about orders of elements in $S_n$ and I know that in order to find max order in $S_n$ we can use Landau function though I think that for small $n$ it is better ...
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1answer
77 views

Which of the following numbers can be orders of a permutation $\sigma$ of $11$ symbols

Which of the following numbers can be orders of a permutation $\sigma$ of $11$ symbols, such that $\sigma$ does not fix any symbols? $1. \;18$ $2.\; 30$ $3.\;15$ $4.\; 28$ could any one just give ...
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1answer
134 views

Getting generating functions for a weirdly-defined weight function

Let $n\in\mathbb{N}$. For a permutation $\sigma:[n]\rightarrow[n]$, we ue the notation $(\sigma(1)\sigma(2)\cdots\sigma(n))$ to describe the mapping. A pair of integers $(i,j)$ is called an inversion ...
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0answers
351 views

Number of permutations such that no two adjacent elements in the original remain adjacent

$N$ students are standing in a line. How many permutations exist such that no two students who were originally next to each other remain next to each other? Suppose $n=4$ and assuming the original ...
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1answer
71 views

Whether solutions of a particular matrix equation are only the permutation matrices

Let $V \in R^{d \times d} $ be a real orthogonal matrix. Denote by $V \circ V$ the Hadamard product (elementwise product). I wish to show that if $V \left(V \circ V \right)^T V$ is "close" to $V ...
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1answer
73 views

Help with Combination

Guys need help to solve this one.. How will we arrange Red balls in '$N$' places , so that if you choose any '$M$' consecutive places, there should be at least '$K$' Red balls among this '$M$' chosen ...
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2answers
436 views

There are 7 empty seats on a bus and four people get on. How many different ways can they be seated?

There are 7 empty seats on a bus and four people get on. How many different ways can they be seated? Would it be 7 pick 4 or 7P4?
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38 views

circle eliminator theory

what is the formula for that, some peoples are standing in a circle remove every second person that in the end only single person remains for example if 5 persons are in circle than first 2nd and ...
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1answer
129 views

$G$ is semiregular implies its centralizer is transitive

How do I prove that the centralizer of every semiregular group is transitive? This is Exercise 4.5 in [Wielandt, Finite Permutation Groups]. Recall that a permutation group $G \le S^\Omega$ is ...
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1answer
82 views

permutation with restriction - no two zeroes can be together from an set consisting of 3 zeroes

I have this as data set - 0 1 0 1 0 now how to find out the number of permutations with restriction that no two zeroes can be together. i.e. 00110, 00011, 11000 ...
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2answers
110 views

Listing subgroups of a group

I made a program to list all the subgroups of any group and I came up with satisfactory result for $\operatorname{Symmetric Group}[3]$ as ...
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1answer
69 views

Arrangement of objects in 2 intersecting circles

In how many ways can 8 tennis balls and 8 baseballs be arranged in 2 intersecting circles (much like on the perimeter of a Venn diagram), if there is a ball placed at each of the 2 intersections of ...
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1answer
3k views

arrangement of objects in circle (circular permutation)

I know circular arrangement of $n$ different objects can be done is $(n-1)!$ ways. For example :- I arranged $7$ objects in circle This can be done in $720$ ways (using $6!$) $1$) Can I also do ...
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1answer
62 views

Selecting a representative permutation

If we have a complete set of permutations {m} = n choose k, how do I select a representative set of permutations in a stream such that the selected set say, {s} keeps growing to include permutations ...
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1answer
148 views

Degree of transitive constituents is odd implies $|G|$ is odd

I want to prove that: the order of a permutation group $G \le S^\Omega$ is odd if and only if the degrees of all transitive constituents of $G$ and the degrees of all transitive constituents of each ...
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3answers
310 views

formula of pascal's triangle

I want to make a program for pascal's triangle,I was reading through the details and found something like this: ...
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1answer
119 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 ...
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0answers
37 views

Calculating the probabilities of different lengths of repetitions of numbers of length 6

This question is similar to the question I asked here: Calculating the probabilities of different lengths of repetitions of numbers of length 4 except now I'm having problem with numbers of length 6. ...
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1answer
88 views

Action of $S_7$ on the set of $3$-subsets of $\Omega$

Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem: Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
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1answer
118 views

Permutation combination problem

This is how Edward’s Lotteries work. First, 9 different numbers are selected. Tickets with exactly 6 of the 9 numbers randomly selected are printed such that no two tickets have the same set of ...
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2answers
1k views

Product of permutation matrices

I want to prove that the product of two permutation matrices is itself a permutation matrix. But I don't know how. Please help!
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1answer
46 views

What is the professional term for the combination of the selection in n out of the total m elements?

I know the number of combinations is called ${}_nC_r$, but what about all the exact outcomes? For example: I have $3$ elements $a,b,c$ and for the parameter $2$, I will have outcomes $$ab,\quad ...
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1answer
190 views

Combination of arrangement and probability

Four guys and four girls are arranged in a row such that no two girls are together. What is the probability that any two of the four guys are together?
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possible combinations of 3-digit

How many possible combinations can a 3-digit safe code have? Because there are 10 digits and we have to choice 3 digits from this, then we may get $10^P3$ but A author used the formula $n^r$, why is ...
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3answers
81 views

permutation/combination problem

There are 3 doors to a lecture room. In how many ways can a lecturer enter the room from one door and leave from another door? I have done like this: They way of entering is 3 and exiting is also ...
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1answer
78 views

Is there a name for this given type of matrix?

Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$? (The motivation for this ...
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2answers
435 views

Calculating permutations if the sequences have to be in ascending order?

How would you go about calculating the number of permutations in ascending order. Obviously if you had (a set of) 3 numbers you have $ 3! $ permutations: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), ...
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1answer
210 views

Grouping natural numbers into arithmetic progression

I need to find the number of ways of dividing the first 12 natural numbers into 3 equal groups (4 numbers each), so that the numbers in any particular group can be arranged in AP (Arithmetic ...
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1answer
90 views

Does the kernel need to be the full automorphism group of the induced subgraph?

Let $\Gamma$ be a simple graph. Suppose its automorphism group $G=\text{Aut}\Gamma$ is imprimitive on its vertex set $V$. Take a block system $\mathcal{B}$ of $V$. Let $B\in\mathcal B$, and let ...