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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 question: Four seats for 14 people, how many ways?

A club has 6 female and 8 male members. A president, VP, secretary and treasurer. In how many ways is this possible if?... a) an equal number of men and women hold office? b) the president or VP is male? I have tried counting both equal quantities…
Jinzu
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permutation question restricting a specific order of elements

Q:- In how many ways can 3 men and their wives be made stand in a line such that none of the 3 men stand in a position that is ahead of his wife? I solved it using a longer and tedious method. By the way i may also know the shorter method but i m…
PleaseHelp
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A question regarding a permutation in $S_n$

Let $\sigma\in S_n$. For each $i\in\{1,2,\ldots,n\}$ let $k_i$ be the smallest positive integer such that $\sigma^{k_i}(i)=i$. Suppose now that $k_1,\ldots,k_n$ are all even. Is it true that $n$ must be even too? My attempt: Suppose that…
boaz
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Criteria to distinguish permutations from the other bijective (&etc.) maps on binary sequences

A follow-up to MSE4856267, some context is here. Consider binary (0/1) sequences of length $n$. Permutations (symmetric group $S_n$ ) act on any sequences - just permuting the symbols. Moreover there are obvious orbits - sequences containing…
Alexander Chervov
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Is the map "NOT" on binary sequence of length $2n$ with precisely $n$ symbols "1" induced by natural action of the symmetric group $S_{2n}$

Consider binary (0/1) sequences of length $2n$. The permutations (symmetric group $S_{2n}$) naturally acts on any sequences on length $2n$ - just permuting the symbols. That action satisfies the condition that $x$ and $p(x)$ contain exactly the same…
Alexander Chervov
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Checking the decision of permutation problem

$$\sigma \in S_n. \ Prove \ that \ \sigma = \alpha \beta, \ if \ \alpha ,\beta \in S_n \ and \ \alpha^2 = \beta^2 = e.$$ Here is my solution: Let's prove that statement by induction, basis ($n=2$) - evident statement. The induction step: the…
Arthur_Kitsuragi
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Permutation of word DISKRET without SIK, DIS and RET

I've got the question of how many words can be made from DISKRET without the new word containing SIK, DIS or RET. For example DSIKRTE is not allowed. My calclation is 7! - 7!/3!-5!-5!-5! = 4560 I have been given the answer 4692 and can't figure out…
user333750
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Distance in bit-reversal permutation

I consider a bit-reversal permutation of interval $[0, 2^k)$. That is, x is mapped to rev(x) (let's assume that binary representation uses k bits, not more). That is, $x = \sum 2^i$ and $rev(x) = \sum 2^{k-i}$. I would like to prove something like…
rbtrht
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Question about permutation and bijection

I know that for a given $\sigma \in \mathfrak{S}_n$, we have $$ \displaystyle \varepsilon (\sigma )=\prod _{1\leq i
Atmos
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Can this framework be looked at as a "number of permutations" problem?

So there is a system (it's actually from a videogame but I will just explain it in simple words) where you need to perform 30 attempts in a row of getting a success with a certain probability. It works like this - you press the button and there is a…
Uncle True
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can we call $\pi(e)$ the rotation of $e$ by permutation $\pi$?

If $\pi$ is a cyclic permutation of a set, and $e$ an element of that set, can we say that $\pi(e)$ is the rotation of $e$ by $pi$? Or is there a better word phrase for it? Take, for example, $\pi=(abcde)$. Then $\pi(e)=a$, and $a$ is the rotation…
Pierre Carbonnelle
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Permutations and combinat

Computer password with 8 characters, but no more than 12 characters, where each character in the password is a lowercase, an uppercase letter, a digit, or one of the 6 special character. How many different passwords are available for this computer…
Kelera
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Calculate the coupling $ \tau^\sigma $

Suppose $ \tau = (a_1, ..., a_k) $ it's a cycle in the group $ S_n $, and $ \sigma $ is any permutation from $ S_n $. Calculate the coupling $ \tau^\sigma $ Could you help me?
Mat
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Permutations by rearranging in groups of 3

Take a deck of cards, cut it in three, and rearrange the three packs in any order (for instance, the middle goes on top, the top in the middle, and the bottom stays). Repeat as many times as you want with different (and possibly uneven) cuts. What…
Arthur B.
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