# Tagged Questions

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

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### Permutation of the alphabets of the word “mediterranean” such that first and fourth letter are “r” and “e” respectively.

Above is the original question. The correct answer is in green that is 59. I have chosen option 3 that is $\frac{11!}{(2!)^3}$ because I thought that there are 13 alphabets in the word "mediterranean"...
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### How can we compute restrictions from a character table?

I would like to how to, when given a character table, calculate the restriction. $Res_H^G : Rep(G) \rightarrow Rep(H)$. For example: Let $G=S_4$ whose character table is given below (see ...
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### Need help calculating number of possible passwords with given criteria

I need help calculating the number of possible passwords with a given set of criteria. Here is the set of criteria: Passwords are case insensitive. Must be 6-14 characters. Must contain at least 1 ...
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### Counting monomials with $k$ variables

Say we expand $\left(\sum_{i=1}^n x_i\right)^k$ into monomials. If $k=3$ there are $3n(n-1)$ monomials with two variables: $3x_1x_2^2 + 3x_1x_3^2 +\dots + 3x_1^2x_2 + \dots$. Is there a closed form ...
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### Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$.

$S_n$ is the set of all permutations. Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$ I can somewhat see why this statement would be true, seeing as permutations are read from right to ...
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### Suppose $π ∈ S_n$, and for this $π$ define $C_π : S_n → S_n$ be defined by $C_π(σ) = πσ$. Why is $C_π$ a bijection?

$S_n$ is the set of all permutations. I'm just starting on this material, so I'm confused on how to read this problem. Does the function consist of multiple permutations (i.e. the permutation of a ...
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### Find The Number Of Outcomes

I understand how to find the number of outcomes using permutations and combinations, but then I thought to myself what happens when it involves both? I will make a mock scenario to explain what I am ...
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### the length of the conjugate class containing $\alpha$ in $S_n$ [duplicate]

Suppose $\alpha$ $\in$ $S_n$ and there are exactly $n_i$ $l_i$-cycles ($i=1,2, ... ,k)$ (containing $1$-cycles) in the cycle decompostion of $\alpha$ ., then the length of the conjugate class ...
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### Converting Permutations to Combinations: Simple Stats in Practise

In a popular text book there is a question that has bothered me that I am sure is very simple for others and I'm just missing something..... So image $100$ songs and we have $10$ as Beatles songs. We ...
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### Generate a unique combination from an index within the number of combinations

I'm writing a program which will use a genetic algorithm optimize neural networks to play tic-tac-toe (That's not related), and I've come across the following problem: I'm looping through every ...
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### Permutation : Is there any formula to solve this?

Given, 14 objects of type A 8 objects of type B 3 objects of type C 2 objects of type D Find the permutation of 10 objects? Is there any general formula in permutation to solve a problem ...
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### What is an intuitive explanation of the combinations formula?

I perfectly understand the permutations formula i.e. if you have $n$ things how many ways can you rearrange it if taken $k$ at a time (or if you have $k$ slots)? So you draw the following tree. And ...
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### Is $N_{A_7}(H) = H$, with the following $H$?

I am following a proof in which I have a subgroup of $S_7$ defined by $H := \langle (2, 3, 4)(5, 6, 7) , (2, 7, 6, 3)(4, 5) \rangle$ The book implicitly uses that $N_{A_7}(H) = H$ (the normalizer ...
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### Removing the dimension factor in Fannes inequality

Given two distributions $x=(x_1,\ldots, x_n),y=(y_1,\ldots y_n)$ on $[n]$, it is known by Fannes inequality that $H(x)-H(y)\leq O(\|x-y\|_1\log n)$, where $H(\cdot)$ and $\|\cdot\|_1$ represent ...
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### Derangements One to One Functions

What is the number of one to one functions from $\{1,2,3,\cdots,n\}$ to $\{1,2,3,\cdots,n\}$ so that $f(x)\neq x$ for all $x$. I understand that $A_1 \cup A_2 \cup A_3 \cup\cdots\cup A_N$ is the set ...
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### Decomposition of permutation

I was asked to decompose the permutation $$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1 \\ \end{pmatrix} = (12345) \in S_5$$ into a product of two ...
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### Polynomial roots conditions vary with coefficients

Polynomial equation $\sum_{i=0}^4 p_i x^i=0$ have the following root conditions: 1) $a_0 \pm b_0 i, a_1 \pm b_1i$ 2) $a_0 \pm b_0 i, a_1, a_2$ 3) $a_0, a_1, a_2 \pm b_2i$ 4) $a_0, a_1, a_2, a_3$ I'm ...
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### What is an intuition behind conjugate permutations?

I know the definition of conjugate permutations. $$\exists p \quad p^{-1} \alpha p=\beta$$ So the $\alpha$ and $\beta$ is a pair of conjugate permutations. But can anybody can give some concise, ...
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### how to find digits except trailing zeroes

I have came across many questions of permutations and combination but I am confused in these types of questions “how to find last two digits" except trailing zeros in $1000!$ where $!$ represents ...
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### Solvability if two pieces of the fifteen puzzle are identical?

It's known that only half of all the permutations in the fifteen puzzle can be solved (in the sense of recovering the sequential order of numbers, with the empty slot in the lower right corner), for ...
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### 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 ...
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### 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 ...
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### 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 $n$...
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### 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 ...
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### 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 ...