Tagged Questions

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

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Random Sequence of Alternating Increase/Decrease Numbers

The problem statement: Repeatedly pick a random number (uniformly-distributed) between $0$ and $1$. Keeping going while the second number is smaller than the first, the third number is larger than the ...
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Counting nearly-sorted permutations

Let $[n]$ denote the set $\{1,2,\ldots,n\}$. We call a permutation $\sigma:[n]\to[n]$, $(n,k$)-nearly sorted if $$\forall i\in [n]: |\sigma(i) - i|\le k,$$ i.e., every element is shifted at most $k$...
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large permutation question

What are the permutations of the following: 7 marbles each of 4 colors, for a total of 28 marbles. A 5x5 board, so 25 places for 1 marble to be placed. What are the permutations of placing the 25 ...
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A problem relating to combinatorics/ Permutation|Combination [closed]

The question is: (I actually may have messed up!) Suppose we have $4$ numbers namely : $a,b,c$ and $d$ Now we need to compare two of these numbers four at a time. For example : $a>b$ |...
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If $\alpha\in S_{k+l}$, $\alpha=\left(\begin{array}{cccccc}1&\cdots&k&k+1&\cdots&k+l\\l+1&\cdots&l+k&1&\cdots&l\end{array}\right)$, for $k,l\in\mathbb{Z}^+$ ...
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Compute Square of Permutation

Dr. Pinter's "A Book of Abstract Algebra" presents the preface to a few exercises: In $S_{5}$, express each of the following as the square of a cycle (that is, express $\alpha^{2}$ where $\alpha$ is ...
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Oberwolfach Problem - 30 people at dinner on 3 tables of 10 seats each

There are $30$ people at an alumni dinner, seated at $3$ round tables of $10$ seats each. After every time interval $\Delta t$, a position change event is required where everyone changes position ...
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Representing a 5-cycle as a product of transpositions

Dr. Pinter's "A Book of Abstract Algebra" shows that: $$(12345)$$ can be written as the following product of transpositions: $$(54)(53)(52)(51)$$ How can the first representation, $(12345)$, be ...
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On permutation notation

I am trying to write up a proof that the parity of permutations on finitely many letters is well-defined. I think I have a proof that involves the number of disjoint cycles that a permutation may be ...
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Identity element of a group as a factorization of group elements.

For any group $G$ we readily verify that if $a,b,c\in G$ and $a*b*c=e$ , where $e$ denotes the identity element,then also: $b*c*a=e$ Indeed,let $b*c=x$.Then our problem amounts to: $a*x=e⇒x*a=e$ This ...
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Split a number into parts

In how many ways can a natural number $n$ be split into $m$ natural numbers (parts) where each part is less than $n$, the parts don't necessarily have to be equal, and all of them add up to $n$?
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seating arrangements of four men and three women around a circular table

In how many ways can $4$ men and $3$ women be arranged at a round tale if i)the women always sit together? ii)the women never sit together? I attempted both the questions but the answers I got don't ...
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A combinatorial question about outer automorphisms of $S_6$

Quite possibly I'll solve this and post my answer below, but maybe others will post better answers before I get to that. Or after.$^\dagger$ The group of permuations of $\{a,b,c,d,e,f\}$ is ...