# Questions tagged [permutations]

For questions related to permutations, which can be viewed as re-ordering a collection of objects.

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### Represent a permutation as a product of disjoint permutation cycles.

$\pi_1=\pmatrix{1 & 2 & 3 & 4 & 5 \\ 5 & 3 & 1 & 4 & 2}$ and $\pi_2=\pmatrix{1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 1 & 2 & 3}$. Let $\pi_3$ ...
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### Normal closure of the Weyl subgroups [closed]

What is the normal closure of the Weyl subgroup in the general linear group GL(n,k)?
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We have 2 strings $|v\rangle$ and $\langle u|$, $|v\rangle=|e_{1}\rangle^{np}|e_{2}\rangle^{n(1-p)}$ where $e_{1}$ occurs $np$ times and $e_{2}$ occurs $n(1-p)$ times and $\langle u|=\langle f_{1}|^{... 1answer 16 views ### Prove that a cycle of length$k\geq 2$can be written as a product of$k-1$transpositions. Prove that a cycle of length$k\geq 2$can be written as a product of$k-1$transpositions as follows: $$(a_1 ... a_{k-1} a_{k})=(a_1 a_{k})(a_1 a_{k-1})...(a_1 a_2).$$ I found an answer here: ... 1answer 24 views ### is this the correct way to solve the question? There are 35 students in David's homeroom class. There are 5 students who take English and Biology,and 7 students who take neither of these subjects.There are 3 more students taking English only than ... 1answer 39 views ### Let us subdivide$n$persons into$m$teams. How often we must rebuild the teams that everyone worked at least once with everyone else in a team? We need to subdivide$n$persons into$m$teams of equal size ($m\mid n$). How often we must rebuild the teams (by shuffling the team members / reassigning them to a different team) so that everyone ... 0answers 13 views ### Probabilites with non repeating numbers I am trying to calculate the likelihood of picking x numbers from a selection out of y whe order is important. I have the following for working out the when the numbers can be repeated, for example ... 1answer 25 views ### Arrangements and permutation How many arrangements of$A, B, C, D, E, F, G, H$are there such that$A$and B are between C and D ? Attempt : I am trying to solve this problem using simple counting process like first place can ... 1answer 51 views ### How many possible possible passwords are there for 8-100 characters? Requirements/Restrictions: Minimum of$8$. Maximum of$100$. At Least$1$letter from the latin alphabet (capitalisation doesn't matter-g is same as G,$26$letters), atleast$1$number ($0-9$,$10$... 0answers 89 views ### Stabilizer and orbit of a polynomial. I was reading permutation part where I found this problem.Let there be a polynomial group$S_4$where$p=x^2_1x_2+x_2^2x_3+x_3^2x_4+x_4^2x_1$with variable$x_1,x_2,x_3,x_4$. Find the stabilizer and ... 1answer 24 views ### Combinatorics - how many ways to divide balls in two groups Suppose I have: 8 black balls 3 white balls 5 blue balls how many ways there are to divide those balls into two different groups (note that there is no need to divide into two groups with even ... 1answer 43 views ### Closed form of$\sum_{n=1}^{\infty} (n-1)!x^n$[duplicate] In a problem that I am trying to solve using generating function, the right-hand side (RHS) of the generating function equation is$\sum_{n=1}^{\infty} (n-1)!x^n$. Would like to find the closed ... 0answers 21 views ### How many arrangements are there of the word “PERMUTATIONS” so that no three vowels come together? I did this- Total Arrangements without any constraint - Total arrangements when 3 Vowels together = Total arrangements when no 3 vowels are together.$\frac{12!}{2} - (\frac{10!}{2} \cdot {}_5C_3 \...
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Been trying to do this question since 3 days yet not been able to approach please!!! help If we disable and technically block all easily guessable weak passwords (specific combinations of letters, ...
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### Circle permutation without consecutive integers placed together

How many ways can $n$ numbers from 1 to $n$ be arranged in a circular order without consecutive integers being placed together? (Note: 1 and $n$ are also considered consecutive integers) For example, ...
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### Cards Shuffle problem: How to prove solution exists? Is there a formula for the solution?

Problem You have a pile of N cards sorted from 1 to N where card 1 is the one on top and card N is the one at the bottom. We do a shuffle operation on the N cards and the shuffle consists of going ...
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### is there any way to find the maximal length of cycle in a permutation group?

How we can find the biggest cycle of a permutation group with given the set of generators ? is there any algorithm or theorem for that? Or how we can find the set of generators with biggest cycle as ...
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### A permutation π on $[n]$ is said to be even-dominated if $\phi_{2i−1}< \phi_{2i}> \phi_{2i+1} \ for \ all 1 ≤ i < n/2$

Let a be the number of even-dominated permutations on $[n]$. Let $a(x)$ be the exponential generating the function of $(a_n)_{n≥0}$.
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