Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [permutations]

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

The word permutation has several possible meanings, based on context. In combinatorics, a permutation is generally taken to be a sequence containing every element from a finite set exactly once. Permutations of a finite set can be thought of as exactly the ways in which the elements of the set can be ordered.

In group theory, a permutation of a (not necessarily finite) set $S$ is a bijection $\sigma : S \to S$. The set of all permutations of $S$ forms a group under composition, called the symmetric group on $S$.

Reference: Permutation.

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How many ways can four letters abcd be arranged such that a always comes before b and c always comes before d?

How many ways can four letters abcd be arranged such that a always comes before b and c always comes before d? Total number of ways abcd can be arranged? 4! Half of them a if before b, half b is before a. Similarly, in half, c is before d, and…
Saad Rehman Shah
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Combinations and Permutations: 3 letters, repeat exactly three times

If I have three letters, A, B, and C, and I want to use each letter exactly three times (9 places), what is the probability that I randomly pull out a nine-string of letters that starts with ABC? I need help figuring out How to find out how many…
danielle
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What is the minimal cardinality for a generating set of the permutations?

I want to find the minimum number of permutations so that all other permutations can be obtained by multiplying the permutations of this set (taken in any quantity). In other words, I am looking for the minmal cardinality of a generating set. I…
Lex
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Cards Swapping Problem

The 100 integers from 1 to 100 are each written onto 100 index cards, which are then thoroughly shuffled. To put the cards back into their natural ordering, the only operation allowed is for a pair of cards to be swapped. Is it always possible to…
Aeon
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'Canonical' form of permutations, product of transpositions

I have such 'canonical' form of permutations: $\prod_{i=0}^n (i \ k_i)$, where $i \leq k_i \leq n$. For example, there are all $6$ permutations of $3$ elements. Of course, some transpositions do nothing and can be removed. $(0 \ 0) (1 \ 1) (2 \ 2)…
zaa
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Ordering letters in the word

Given word MISSISSIPPI I am asked to count the number of (1) permutations where all P's are side-by-side (2) permutations where P's are side-by-side and I's are side-by-side. For the first question, I did $$\frac{9!}{4! 4!} \times 10$$ since I have…
permute
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Is it possible to arrange 13 counters on a 5x5 grid such that all 2x2 squares are unique?

Given a 5x5 grid of 25 tiles, is it possible to place 13 counters so that no 2x2 internal square of the grid is repeated? Each square can be considered binary, with either a counter or no counter. Here is an example of a configuration of 13 counters…
Phoebe J
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Determine if number of permutations is odd or even

I am facing problem in this question. Let $X$ be the total number of $n$-permutations that have exactly $k$-inversions. An inversion is defined as a pair of entries $(i,j)$ such that $i p(j)$. Determine whether $X$ is odd or…
Jfk
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How many ways to build a hamburger?

Friendly's recent advertising campaign claims there are over 10 trillion combinations of hamburgers. Their options are: Protein (4) Bread (5) Cheese (7) Premium topping (3) Hot toppings (10) Cold toppings (8) Sauces (11) Their calculation is…
Dan
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Calculating the sign of the generalised permutation

What is the sign of the following permutation. Prove your answer: $$\pmatrix{1 & 2 & \cdots&p&p+1&\cdots & \cdots &p+q \\ q+1 & \cdots & \cdots & q + p& 1 & \cdots& \cdots &q}.$$ I said let $\tau$ equal the permuation. Then, we get sign$(\tau) =…
Kaish
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Maximum of $S(\sigma) =\sigma(1)\sigma(2)+\sigma(3)\sigma(4) + \cdots +\sigma(n-1)\sigma(n)$ where $\sigma \in S_n$

I want to calculate the maximum for $\displaystyle S(\sigma) = \sigma(1)\sigma(2)+\sigma(3)\sigma(4) + \cdots +\sigma(n-1)\sigma(n)$ where $\sigma \in S_n(n:even)$. I tried brute force for small $n$. When $n=2$, any $\sigma \in S_2$ satisfies…
Kaira
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Find all solutions in an equation with permutations in $S_{10}$

Let $\sigma=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 2 & 9 & 5 & 7 & 10 & 3 & 4 & 6 & 1 & 8\end{pmatrix} \in S_{10}.$ Find all permutations $\tau \in S_{10}$ where $\tau^3 = \sigma.$ My first intuition was to multiply the…
user638287
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How to identify whether permutation question involves repetition or not

1) How many three-digit numbers more than 600 can be formed by using the digits $2,3,4,6,7$? 2) How many number greater than a million can be formed with the digits $2,3,0,3,4,2,3$? How could I distinguish between a repetition and non repetition…
J.Doe
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How many combination can we create from this structure?

I 'm stuck in to this problem and I really need your help. Sorry if the title is not very informative. It is really hard for me to explain it in one sentence. I do my best in explaining it: There exists these two natural integers: m $\in…
Ramin
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Alternative definition of the sign of a permutation

In a groups course in my first year of university we defined the sign of a permutation by showing that every permutation can be written as a product of transpositions and defining the sign of the permutation as $(-1)^n$ with $n$ the number of…
Asier Calbet
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