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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 to compute length of permutation in symmetric group

What is the length of a permutation and how do you compute it? For example what are the permutations in S_4 of length 1?
jfcjohn
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Index of a permutation of N natural numbers

Suppose I have a random permutation of first $N$ natural number, how do I determine its index of a given permutation, if all the permutations were arranged in lexographical order , I understand the approach for $N<10$ as all numbers are single…
user2970016
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Permutations and fundamental principle of counting.

find the number of different 8 letter arrangements that can be made from the letters of the word DAUGHTER (8 letters) so that --All vowels occur together. I thought of a solution which is as follows-- if all the vowels are occurring together then…
Freelancer
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The number of sequences in which 7 players can throw a ball,so that the youngest player may not be the last ball is :

The number of sequences in which 7 players can throw a ball,so that the youngest player may not be the last ball is : a) $4000$ b) $2160$ c)$4320$ d) $5300$ MyApproach: The first 6 players can throw a ball in 7! ways and the last player can throw in…
Jack
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A man has (2n+1) friends.The number of ways in which he can invite atleast n+1 friends for dinner is 4096.Find the number of friends of the man

A man has (2n+1) friends.The number of ways in which he can invite atleast n+1 friends for dinner is 4096.Find the number of friends of the man Options given: 1) $11$ 2) $13$ 3) $15$ 4) $17$ My Approach: I did $2$^$(2n+1)$ -$1$=$2$^$12$ I am not…
justin takro
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How to find the inverse of a permutation function from its 2-cycle decomposition?

Could I quickly spot the inverse of a permutation from its 2-cycle composition? For example, given that $\rho=(1 \ 9)(1 \ 4)(1 \ 5)(1 \ 8)(2 \ 10)(2 \ 3)(2 \ 6)(2 \ 7)$, how to find its inverse from this 2-cycle decomposition?
Rescy_
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Showing that the $T_n + T_{n+1}$ equals the number of ways of occupying something different

I am having trouble with the wording of this question. $Tn$ is the number of derangements of a set of size $n$. $n$ treeloppers are cutting down $n+1$ trees. They go and have a lunch break in the lodge and return to start chopping the $n+1$ trees…
Dan
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16 teams enter a competition

16 teams enter a competition. They are divided into four pools (A, B, C and D) of four teams each. Every team plays one match against the other teams in its pool. After the pool matches are completed: the winner of pool A plays the 2nd placed team…
atong
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Are cyclic permutations always even?

I just discovered the permutation symbol $\epsilon_{ijk}$ and was wondering if the set of cyclic permutations and even permutations are the same.
Malz
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Number of injective maps from one finite set to another

$m\le n$ be natural numbers. What is the number of injective maps from a set of cardinality $m$ to a set of cardinality $n$ $?$ I think it is the number of ways $m$ distinct elements can be chosen from $n$ elements…
user118494
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If abc is a three digit number such that not one number is similar to other than how many possible values of ( a + 4b + c ) will be divisible by 40?

If abc is a three digit number such that not one number is similar to other than 1 How many possible values of ( a + 4b + c ) will be divisible by 40? 2 How many possible values will be divisible by 40 : My approach: i could solve only second…
Jack
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Permutation as product of inversions

Each permutation in $S_n$ can be written as a product of inversions (not disjoint) $(1 2 3 4 5)=(1 5)(1 4)(1 3)(1 2)$ How do I "read" it? I first start we the right side $(1 2)$?
gbox
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Find the number of ways this can be arranged in which no 2 women and no 2 men sit together given 4 men and 3 women are seated in a dinner table?

Find the number of ways this can be arranged in which no 2 women and no 2 men sit together given 4 men and 3 women are seated in a dinner table? @Edit: They are seating in a row dinner table I have applied 2 approaches in this question and found…
Jack
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In $S_7$, find the permutation $x$ that satisfies the condition $x^2 = (1,2)(3,4)$.

In $S_7$, $X^2 = (1,2)(3,4)$. What is the permutation $X$ that satisfies the given condition?
Nazare Andrei-Cristian
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Ordered Arrangement of Books

A shelf has $10$ distinct books of which $3$ are Math,$ 4$ are English and $3 $ are Science books. In how many ways can you arrange these books such that all English books will be placed in the center? I have tried to answer the arrangement where…
Florencio
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