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

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3
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2answers
121 views

Permutations of numbers $1, 2, 3,\dots,n$

How many permutations do the numbers $1, 2, 3,\dots,n$ have, a) in which there is exactly one occurrence of a number being greater than the adjacent number on the right of it? b) in which there are ...
1
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0answers
12 views

What's uniform block signed permutations?

Let $[n]=\{1,2,\ldots,n\}$ and $P(n)$ the set of all partitions of [n]. A partition of $[n]$ is non-empty disjoint subsets of [n], called blocks, whose union is $[n]$. A block permutation of [n] is ...
0
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1answer
27 views

Filling k positions with objects from $n$ different types

There are $n$ different types of objects and $k$ positions where an object can be placed. How can I determine the number of ways in which these $k$ positions can be filled by using objects of these ...
0
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0answers
14 views

Matlab: How to find a permutation matrices

I'm trying to figure out a way to compute the permutation matrices R and L given two matrices A and B. I would like to get L and R given that I know A and B. B=L* A* R. I wrote the code below for ...
0
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0answers
15 views

Decompose induced representation of $S_2$ and $S_3$

Let $ H=S_2 \subset G=S_3 $. Then use Frobenius reciprocity to decompose $ \operatorname{Ind}_H^G(\operatorname{sgn}_H) $ into irreducibles. $ G=S_3 $ has $ 3 $ irreps $ 1_G, ...
1
vote
1answer
62 views

$10$ people are standing in a queue when three new checkouts open. In how many ways can the three new queues be formed?

Problem: $10$ people are standing in a queue when three new checkouts open. 8 people rush to the new checkouts and the new queues end up with at least two people in each. In how many ways can ...
1
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3answers
37 views

Probability in Permutation [closed]

A permutation of $1,2,3,\ldots,n$ is chosen at random. Then the probability that the numbers $1$ and $2$ appear as neighbours equals ______? Options are A) $\dfrac 1 n$ B) $\dfrac 2 n$ C) ...
0
votes
1answer
29 views

Prove the pullback of the wedge product is the wedge product of the pullbacks.

Let $F:V \rightarrow W$ be a linear map. Show that $F^{\ast}(\omega \wedge \eta)=(F^{\ast}\omega) \wedge (F^{\ast}\eta)$ for all $\omega \in \Lambda^{p}(W) , \eta \in \Lambda^{q}(W)$. Where ...
3
votes
2answers
63 views

math software - permutation group elements operation

I need a software allows to calculate operation elements of permutation group. For example the following elements operation yields identity permutation $$ (1234)(1423) = (1)$$ Sage seems to solve the ...
1
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0answers
27 views

Trouble understanding Sylow's Third Theorem

The statement of Sylow's third theorem in my text goes like this, Let p be a prime and let G be a group of order $p^km$, where $p$ does not divide $m$. Then the number $n$ of Sylow $p$-subgroups of ...
0
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0answers
28 views

Modifying permutation function for inputs with equivalent ratios

I have the following function: $$f(a_1,a_2,\ldots,a_n) = \frac{(a_1 + a_2 + \cdots +a_n)!}{(a_1! a_2! \cdots a_n!)}$$ where $a_i\ge 0$ I need to modify this function such that $f(a_1,a_2,...,a_n) = ...
0
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0answers
22 views

Different ways in which a micro-switch with eight switches can set

A computer interface for a Kawai digital studio piano has eight micro-switches that can be set in either the "on" or "off" position. These switches must be set properly for the interface to work. In ...
-1
votes
2answers
45 views

How to calculate the number of non decreasing functions between two finite sets? [duplicate]

I want to know how to calculate number of non decreasing functions from one set to another set. Let $A=\{1,2,3,\ldots,10\}$ and $B=\{1,2,3,\ldots,25\}$ Please tell me an easy method to calculate the ...
1
vote
2answers
50 views

Bernoulli Numbers and Tangent numbers.

Good evening. I am looking to see if there is a proof online to help guide me with the understanding that the Tangent Numbers, denoted $T_n$ and the Bernoulli numbers, denoted $B_n$ are related. It ...
1
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2answers
73 views

How many permutations can be formed from $2n$ distinguishable objects and $n$ indistinguishable objects?

How many permutations can be formed from $2n$ distinguishable objects and $n$ indistinguishable objects? Please tell me if I am on the right track to solving this question. Basic Formula: ...
-2
votes
1answer
20 views

Generate all Permutations of Four Events, Three Outcomes each

Hello I would like a list of all permutations for the following set up. I tried an online permutation generator, but I didn't quite get it working, so I'll try this forum, which has been great in the ...
0
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0answers
22 views

A probability word problem

There is an infinite rectangular window, with infinite vertical iron bars dividing it at distances of 30 centimetres. Now, a 15 cm long pen is randomly thrown at the window. Assuming that the pen can ...
1
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1answer
33 views

Is an orthogonal matrix necessarily a permutation matrix?

Is an orthogonal matrix necessarily a permutation matrix? I believe the answer is no as a permutation matrix is a special case of an orthogonal matrix, but I am having a trouble finding a ...
0
votes
1answer
14 views

Find the left coset $(13)H$ of $H= \langle (12) \rangle$ in $S_5$

I'm a bit stuck on this problem. I understand that $S_5$ is the group of symmetries with $n=5$, that's trivial. I'm also aware that there are $120$ subgroups in $S_5$ and $2$ in $H=\langle(12)\rangle$ ...
0
votes
1answer
36 views

Permutation of boy and girl names

During your pregnancy, you decided on a list of 23 girls’ first names and 16 boys’ first names, as well as a list of 11 gender-neutral middle names. To your surprise, you had quintuplets, two boys and ...
1
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0answers
21 views

“Tetris permutation” set generation

In the game of tetris you are guaranteed to get each of the 7 unique pieces in some random order. For this we will call them abcdefg. This would give us ...
2
votes
1answer
30 views

order of $(1,3,2,4)(4,3,5)$

Write the $ \pi=(1,3,2,4)(4,3,5) $ as a) product of disjoint cycles b) product of transpositions, and is $\pi$ odd or even? c) order of $\pi$ I think this is right $$(1,3,2,4)(4,3,5)= ...
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0answers
35 views

Is it true in the Group of Symmetries that (ab)(ac) = (acb)?

Suppose $1 \le a,b,c \le n$ and $a\ne b$, $a\ne c$, $b\ne c$. Is it true in $S_n$ that $(ab)(ac)=(acb)$? I'm generally new to the subject of Applied/Abstract Algebra and feel as if this is easier ...
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2answers
29 views

Can you create a formula for the amount of possible permutations of a three digit number, who has a digit sum equal to 4

Can you create a formula for the amount of possible permutations of a three digit number, who has a digit sum equal to n?
0
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0answers
17 views

number of permutation in a boolean expression containing only ANDs and ORs

I need to find the number of permutations of some expression which contains only conjunctions and disjunctions e.g.: $$ e = x_1x_2 \vee x_3x_4 $$ where $x_1x_2$ and $x_3x_4$ are boolean summands, ...
5
votes
3answers
60 views

Number of subsets from an ordered set where adjacent elements may or may not be tied together

Assume we have an ordered set $S$ with a finite number of elements $S=\{1,2,3,\ldots,N\}$. I need to know the number of subsets where adjacent elements from the original set may either be tied ...
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0answers
25 views

Some properties of finite group of order $p^aq^b$

Let $G$ be a finite group of order $p^aq^b$ ( $p$, $q$ are two distinct primes and $a, b\geq 1$) with $\operatorname{Z}(G)=1$ and $P\in \operatorname{Syl}_p(G)$, $Q\in \operatorname{Syl}_q(G)$. Also ...
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0answers
48 views

Enumerating strings with permutations

A string of length n is called valid if it does not contain three indices $i,j,k$ such that $str[i]$$=$$str[j]$$=$$str[k]$ and $2$$*$$j$$=$$i$$+$$k$. The question is to calculate the number of valid ...
0
votes
1answer
52 views

Natural numbers in a circle, combinatorics, existence

I need help with a problem whose solution I'm unaware of. The first $74$ natural numbers are arrange in some manner in a circle. Does there exist an arrangement such that every sum of three ...
0
votes
1answer
45 views

Permutation question (Known answer, don't understand how to get to it)

To save your time, I simplify the question to something like this: There are $18$ people, $4$ of which are teachers. All of them($18$) are going to stand in a row. In how many ways can they be ...
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3answers
51 views

In how many ways can a positive integer $n$ be expressed as a summation of positive integers less than $n$

For example if I take $n=5$, then I can express it in the following ways: $1+1+1+1+1$ $2+3$ $3+2$ $1+4$ $4+1$ $1+1+3$ $1+3+1$ $3+1+1$ $2+2+1$ $2+1+2$ $1+2+2$ Please note that the order of terms in ...
0
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0answers
35 views

Find the string corresponding to a particular lexicographical rank

I came across this question few days back- Given a string - “ thereanswerisyetinsufficientmeaningfulasforadata “ , form all the words with atmost 15 letters and arrange them in lexicographical ...
0
votes
1answer
23 views

Series that converge to every real number via permutation

This great answer at MathOverflow, http://mathoverflow.net/a/29488/8784, shows that the set of permutations of $\mathbb N$ is uncountable. However, I did not grasp the fact that he uses: any ...
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0answers
12 views

Ordering of basis elements of a Lie-group representations tensor product

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \bigoplus_J \textbf{r}_J $$ where $\textbf{r}_J$ ...
1
vote
1answer
35 views

How many partitions of $n$ are there?

Considering a partition to be an ordered $n$-tuple $(m_1, m_2, m_3, ..., m_n)$ with all the numbers $m_i$ natural, $m_1 \le m_2 \le m_3 \le ... \le m_n$, and $m_1+m_2+...+m_n=n$: how many of those ...
2
votes
1answer
57 views

Combinatorial proof of an identity between restricted counts of permutations and derangements

In an answer to Counting permutations with given condition, I showed that the number of permutations of $k$ elements that satisfy $\sigma(i+1)\ne\sigma(i)+1$ is $\frac{!(k+1)}k$, which is the number ...
2
votes
1answer
84 views

Circle Permutation w/ Restrictions questions

I was working on two permutation questions, and wasn't sure if I arrived at the correct answer. 1.) In how many ways can a family of four (mother, father, and two children) be seated at a round ...
0
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1answer
29 views

Permutations Question Thinking Question (8 students in circle)

For a game of London Bridge, 8 kindergarten students form a circle holding hands and then walk in a clockwise direction. If the Prefect in charge allows the children to stand wherever they wish, in ...
9
votes
2answers
152 views

Seeking combinatorial or group theoretic proof for permutation identity

While working on another problem, I found the following combinatorial equality, but I got it analytically, and I'm curious to find a counting argument. Fix $n$ a positive integer. For $n_1\leq ...
2
votes
1answer
81 views

Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
2
votes
2answers
39 views

Sitting n families around a circular table with a condition

How many ways are there for sitting n families around a circular table. Each family is a mother a father and a child. Condition: The mother and father of each family should be sitting next to each ...
1
vote
1answer
24 views

Find the number of paths given the probabilities of each move (Probability and Permutation)

A robot is programmed to move on a flat surface one step at a time, either upward (U) or downward (D) or to the left (L) or to the right (R). Each move is independent of the preceding move and the ...
2
votes
1answer
29 views

Meaning of 'there are exactly $2$ letters between any $2$ 'E'' (Permutation and Combination)

In a game show, the host gives an incorrect arrangement of the letters of the word 'EXCELLENT' and lets the contestant guess the word within a given time limit. Find the number of different ways that ...
0
votes
0answers
44 views

Pairs of Numbers such that the sum of their digits is Equal

How many pairs of numbers $(n,m)$ whose digits add up to the same sum, where $n\ne m$ and $(n,m)=(m,n)$ such that $m,n\le k$ , are there for a given $k$? Observing this in base 10 we are looking at ...
0
votes
3answers
75 views

In how many ways can the four walls of a room be painted with three colours so that no two adjacent walls have the same colour?

In how many ways can the four walls of a room be painted with three colours so that no two adjacent walls have the same colour ? I specifically want to use inclusion exclusion principle. So ...
0
votes
3answers
55 views

Is it permutations or combinations to find the number of ways 36 characters can be arranged in a 4-letter sequence?

If I have a number in base $36$ (a to z, 0 to 9), and I want to see how many ways those characters can be arranged in a $4$-digit number, what is that called? Is it a permutation or a combination? ...
0
votes
1answer
39 views

Permutation matrix homomorphism

Can someone please help me prove that permutation matrix is homomorphism? By that, I mean, let $f: S_n \to GL_n (\Bbb R), f(\sigma)=A_\sigma$ is homomorphism. The book tells me to prove it myself I ...
1
vote
1answer
146 views

Minimum number of balls to choose such that $k$ balls are of same color

A bag contains $a$ red balls, $b$ green balls and $c$ blue balls. We can take balls out of bag without knowing which one we choose (blindfolded). We do not replace the balls back in bag, we simply ...
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0answers
25 views

Transitive action of a $p$-group on minimal block systems

I have trouble proving the following theorem: Let $P$ be a transitive $p$-subgroup of ${\rm Sym}(A)$ with $|A| > 1$. Then any minimal $P$-block system consists of exactly $p$ blocks. Furthermore, ...
2
votes
2answers
41 views

Find how many different circular bracelets can be formed using $6n$ blue and $3$ red beads, where $n$ is a positive integer.

Find how many different circular bracelets can be formed using $6n$ blue and $3$ red beads, where $n$ is a positive integer. As these are circular permutations where flipping does not make any ...