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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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Permutation is a Transposition

For $i \in \{1,2,...,n-1\}$ let $\tau_i \in S_n$ be the transposition of $i$ and $i+1$. I want to know wether $$\tau_i \circ ... \circ \tau_2 \circ \tau_1 \circ \tau_2 \circ ... \circ \tau_i $$ is a permutation that is also a Transposition. I guess…
calculatormathematical
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Lexicographical order number for a permutation

I've found in a paper the formula for the lexicographical order of a permutation, but I didn't find anywhere a proof for it. For example, we have the following set $S=\{1, 2,3\}$. We have $3! = 6$ permutations: $$\sigma_1 =(1, 2, 3) \\ \sigma_2 =(1,…
andrei81
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Permutation of sequences

Suppose $X$ is the sequence of zeroes and ones of length $m$. Let $x_i\in \{0,1\}$ for $1\leq i \leq m$. $$X=\{x_1,x_2,\ldots, x_m\}$$ Similarly let $Y$ be such a sequence of length $m$ such that the number of nonzero entries of $X$ and $Y$ are…
blackbeard
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arrangement of objects in circle (circular permutation)

I know circular arrangement of $n$ different objects can be done is $(n-1)!$ ways. For example :- I arranged $7$ objects in circle This can be done in $720$ ways (using $6!$) $1$) Can I also do this problem as below ?? I made circular arrangement…
rst
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in how many ways can 8 married couples dance a circular dance

a. with no other restrictions 16!/16 = 15! b. every married couple dances together 7! for the couples and 2^8 for alternating husband and wife. 7!*2^8=1296240 c. exactly 3 couples dance together (the wife dances beside the husband). I believe…
gav
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Is there an easier method to solve this question on permutation?

Question: Calculate the largest possible order of any permutation in S10. My approach: List out all possible order in S10 then find the LCM of each, got lcm(5,3,2) = 30 as the largest possible order. I'm just wondering if there's a quicker method to…
sasaki085
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Find original permutation based on intersection sets

Consider the permutation graph below: Let's say I only have the amount of numbers in the permutation. 5, in this case, and all intersections between the first and the second parallel lines, like {2,5},{2,4}, and {2,3}, for example. With only this…
Lucas Meine
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Is it possible to divide n in d parts where each part can have a specific range of values?

Given a value n and number of parts d. Each part has max and min values it can have. Is it possible to divide n in d parts fullfilling the max and min value criteria ? Example d=2 n=5 Part 1 range-> 0 to 1 Part 2 range -> 3 to 5. Answer is YES…
Ashesh Vidyut
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Diagram 9, shows a four-digit passcode '0131' set by San on his smartphone. How many different passcodes can be formed?

Diagram 9, shows a four-digit passcode '0131' set by San on his smartphone. Diagram 9 He wants to reset the passcode such that the new passcode cannot consists of digit 1 followed by digit 3. How many different passcodes can be formed? I need your…
my acc
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problem in permutation question.

Find the number of hexadecimal numbers containing at maximum 16 hexadecimal digits with all of the digits 0,1, and A present at least once? Give your answer as a hexadecimal number.
Aayush Kalra
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$8$-digit numbers that can be formed by using all the digits $0,1,2,3,4,5,7,9$

If the total number of ways in which $8$-digit numbers can be formed by using all the digits $0,1,2,3,4,5,7,9$ such that no two even digits appear together is $(5!)k$, then $k$ is equal to? There are three even digits ($0,2,4$) and five odd digits…
aarbee
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Prove that sgn$(\sigma \tau) = $sgn$(\sigma)$sgn$(\tau)$

Prove that sign$(\sigma \tau)$ = sign$(\sigma)$sign$(\tau)$ for any permutations $\sigma, \tau \in S_n$. I think the two thing's I'm trying to show are: If sign$(\sigma)$ = sign$(\tau) = \pm 1 \implies$ sign$(\sigma \tau)$ = $1$ Wlog, if…
Kaish
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Probability of new seating arrangement given following conditions

A small class of nine boys are to change their seating arrangement by drawing their new seat numbers from a box. After the seat change, what is the probability that there is only one pair of boys who have switched seats with each other and only…
SolvingTrainee
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Complex Permutations

I am a middle school math teacher helping my niece with permutations. Her teacher gave her a problem and later gave her the solution to the problem. I do not agree with the way her teacher solved it and I would like guidance. Here is the…
Z. Andrea Durio
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What's it called when you generate all permutations with replacement for a certain size and is there a formula to calculate the count?

If I have a set of numbers $\{1,2\}$ the permutations are $\{1,2\}$ and $\{2,1\}$. We can calculate the number of permutations using $\dfrac{n!}{(n-r)!} = \dfrac{2!}{0!} = 2$. However, if I instead say I want to find all permutations of the set…
friartuck
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