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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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Inversions of a permutation. Confused

Sorry for this basic question. In here, we have $2$ inversions of $1$ element (from the set $\lbrace 1,2,3\rbrace$): $$ 132, \\ 213, $$ and that $321$ is a $3$-element inversion permutation. Why $321$ is not a 1-element inversion (just switch $1…
user2820579
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Permutations of $n$ objects where $r = n -1$

In my text book the question is as follows: Find the way in which $5$ persons can sit in a row if two insist on sitting next to each other. They give the answer as $48$. I fail to understand how they got there because when I try the standard…
user337046
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Write $π = (3, 2, 5)(2, 5, 4)$ in “table” notation?

Isn't this impossible...? Because this permutation goes from 3 --> 2 ---> 5 ---> 3 according to the first cycle, but goes from 2 --> 5 ---> 4 ---> 2 according to the second cycle. So 5 can't go to 3 and 4.
Chris
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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" and now we are fixing two alphabets so we are…
Singh
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Show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$

Here's the full question: If $σ ∈ S_n$ is any permutation and $i_1, . . . , i_k $ are $k$ distinct elements of $\{1, . . . , n\}$, show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$ I was given this hint: For $j ∈ \{1, . .…
Chris
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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 factorial sign any ideas would be appreciable.
ashi
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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: $C(n+r-1,n-1)$. $r = n$ $n = 2n$ $C((2n+n-1),(2n-1))$ so…
Sai N
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"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 7! or 5040 unique starts. However, there is also a "hold feature", which will swap the current…
John Howard
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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 the expression of summation also counts if the…
ThePhysicist
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Number of permutations with limited number of repetitions

How many four-digit numbers can be formed with the numbers 1, 1, 1, 2, 2, 3, 3, 4 (it means you can use 1 for three times, 2 and 3 two times, and 4 just once)? How does a generalization of that look like? For instance, how many $k$-digit numbers can…
montefuscolo
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Two men, Adam and Charles, and two women, Beth and Diana, sit at a table where there are seven places for them to sit down

Question Two men, Adam and Charles, and two women, Beth and Diana, sit at a table where there are seven places for them to sit down. Two people are sitting next to each other if they occupy consecutive chairs. A non-trivial rotation defines…
Kiran
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Permutations in which $d$ appears before $b$.

Consider all possible permutations of eight distinct elements $a, b, c, d, e, f, g, h$. In how many of them, will $d$ appear before $b$? Note that $d$ and $b$ may not necessarily be consecutive.
Alex_ban
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In how many ways can 4 red balls and 7 blue balls be arranged in 3 boxes

In how many ways can 4 red balls and 7 blue balls be arranged in 3 boxes where each box must contain at least 1 red ball and each box can contain less than or equal or 4 balls, under the following cases? (1) Each ball is distinct and Each box…
Kiran
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Transpositions and the identity permutation

Prove that $\sigma^2$=$\epsilon$ if and only if $\sigma$ can be expressed as a product of disjoint 2-cycles. $\epsilon$ denotes the identity permutation. Any hints to help me get started would be really appreciated!
Alex
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Multiplication of transpositions?

I can't seem to understand how the multiplication of two transpositions yield the results below: $(x b)(x a) = (x a)(a b) \\ (c a)(x a) = (x c)(c a)$ I can't figure it out for the life of me. I'm trying to learn permutation multiplication on my own,…
user312437
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