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 are there for 8 men and 5 women to stand in a line so that no two women stand next to each other?

I have a homework problem in my textbook that has stumped me so far. There is a similar one to it that has not been assigned and has an answer in the back of the textbook. It reads: How many ways are there for $8$ men and $5$ women to stand in a…
Mark
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6-letter permutations in MISSISSIPPI

How many 6-letter permutations can be formed using only the letters of the word, MISSISSIPPI? I understand the trivial case where there are no repeating letters in the word (for arranging smaller words) but I stumble when this isn't satisfied. I'm…
George Zhai
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Why this $\sigma \pi \sigma^{-1}$ keeps appearing in my group theory book? (cycle decomposition)

I'm studying cycle decomposition in group theory. The exercises on my book keep saying things like: Find a permutation $\sigma$ such that $\sigma (123) \sigma^{-1} = (456)$ Prove that there is no permutation $\sigma$ such that $\sigma (1 2)…
Guerlando OCs
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Putting socks and shoes on a spider

A spider needs a sock and a shoe for each of its eight legs. In how many ways can it put on the shoes and socks, if socks must be put on before the shoe? My attempt: If I consider its legs to be indistinguishable, then it's exactly the…
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How to find the square root of a permutation

Observe that $$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 1 & 5 & 3 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 1 & 5 & 3 \end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 5 & 2 & 3 & 1 \end{pmatrix},$$ so…
Ashot
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Intuitive explanation of binomial coefficient

We know that $$\dbinom{n}r = \dfrac{n!}{(n-r)!r!}$$ An intuitive explanation of the formula is that, if I partition the total number of permutations of objects by $r!$, and choose one member of each partition, then no similarly ordered pattern will…
Chris Kim
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Proof that no permutation can be expressed both as the product of an even number of transpositions and as a product of an odd number of transpositions

I am aware that there are a couple of well-known proofs of this theorem, but I'm specifically grappling with the proof given in Fraleigh's A First Course in Abstract Algebra (Theorem 9.15 in the textbook). Let $s$ be a permutation in the symmetric…
ryang
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Number of permutations of $n$ elements where no number $i$ is in position $i$

I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, $3,1,5,2,4$ is an acceptable permutation where $3,1,2,4,5$ is not because 5 is in position 5. I know that the…
Will specht
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Solving Rubik's cube and other permutation puzzles

I've seen two questions on solving the Rubik's cube but none of the answers have given a complete solution using mainly mathematical techniques. Furthermore, I've not seen a good explanation of general techniques for solving permutation puzzles in…
user21820
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$(12)$ and $(123\dots n)$ are generators of $S_n$

Show that $S_n$ is generated by the set $ \{ (12),(123\dots n) \} $. I think I can see why this is true. My general plan is (1) to show that by applying various combinations of these two cycles you can get each transposition, and then (2) to show…
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How many ways are there to fill a 3 × 3 grid with 0s and 1s?

Extra conditions that put a formal solution out of my reach: the centre cell must contain a $0$, and two grids are equal if they have a symmetry, e.g. $$\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 1\end{array} \right) = \left(…
Sup Bro
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Minimal generating set of Rubik's Cube group

The Rubik's Cube group is generated by the six moves $\{F,B,U,D,L,R\}$. However, is this the minimal generating set for the group? In other words, can I simulate the move $F$ just by making the moves $B,U,D,L,R$? If I try this out on an actual…
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Normal subgroups of infinite symmetric group

I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the set $\mathbb{N}$) has exactly 4 normal…
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Why do we divide Permutations to get to Combinations?

I'm having a hard time reasoning through the formula for combinations $\frac{n!}{k!\left(n-k\right)!}$. I understand the reason for the permutations formula and I know that for the combinations we divide by $k!$ to adjust for the repeated cases,…
Daniel Oscar
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Show group of order $4n + 2$ has a subgroup of index 2.

Let $n$ be a positive integer. Show that any group of order $4n + 2$ has a subgroup of index 2. (Hint: Use left regular representation and Cauchy's Theorem to get an odd permutation.) I can easily observe that $\vert G \vert = 2(2n + 1)$ so $2 \mid…
Robert Cardona
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