# Questions tagged [permutation-cycles]

For elementary questions concerning permutation cycles and permutation groups. This includes all representations of permutations (two-line arrays, cycles, bipartite graphs); transpositions and the sign/parity of a permutation; the Symmetric group and the Alternating group. To be used with the (permutations) tag to make the distinction between abstract algebra permutation questions and combinatoric permutation questions.

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### Decreasing probability of finding a number in a randomized array from 1/n to 2/n using permutation graph

This is a question that my professor asked to the class. There are 52 cards with numbers ranging from 1 to 52 on them. There will be 2 players, and their aim is to find a card on the card list in at ...
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### Let $I$ be the collection of all involutions (for all $n \geq 0$). Find the EGF for $I$. [closed]

A permutation $\pi$ of $[n]$ is said to be an involution if its cycle decomposition consists of only $1$- or $2$-cycles. Let $I$ be the collection of all involutions (for all $n \geq 0$). Find the EGF ...
47 views

### Finding a simple graph such that its automorphism group equals the subgroup of $S_3$ generated by a 3-cycle

I have found that the subgroup of $S_3$ generated by a 3-cycle is $\{e,(123),(132)\}$ where $e$ is the identity but I can't find any graphs that have this group as their automorphism group. I am a ...
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1 vote
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### Multiplicative orders modulo divisors of the modulus

Is there a known description of the set of multiplicative orders of a fixed unit $a$ modulo all divisors of some modulus $n$, i.e. of $\text{ord}_d(a)$ with $d\mid n$? It is easy to see that it is a ...
• 11.7k
1 vote
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### Brainfreeze! reduce a number by a percentage so that the number after X iterations is below 1 [closed]

I am having a sort of brainfreeze or cannot express the mathematical formula for the following: I have a Natural Number X and I want to reduce it by a Percentage e.g 4% Y times that after Y iterations ...
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1 vote
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### Algorithm to create a polynomial invariant only under specific permutations of the variables

I was solving the following problem (1.2.10 from Dixon and Mortimer's Permutation Groups): Given the group $G =\langle(x_1,x_2, x_3, x_4),(x_1,x_3) \rangle$, give an example of a polynomial that's ...
• 7,133
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### Find a $6$-cycle $C$ such that $C^3=(2 \; 9) (5 \; 13) (11 \; 12)$

My initial problem was: Given $\sigma$ = $(1 \; 7 \; 3 \; 15) (2 \; 9) (4 \; 10 \; 6 \; 8 \; 14) (5 \; 13) (11 \; 12)$. Find $\tau$ such that $\tau^3 = \sigma$. I know that there are $2$ cases. First ...
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### Cardinality of a certain subset of group $S_k$ defined in connection with cycle decompositions

I am interested in a subset of the permutation group of $k$ elements, $\Sigma_k$. Any element in $\Sigma_k$ can be decomposed into disjoint cycles in a unique way. Conversely, if we take a partition ...
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### Invariants of the Hyperoctahedral group

Apologies in advance for what I am asking might be too trivial, I am not a mathematician. I have a function $V(x_1,\dots,x_n)$ that could have, at most, a hyperoctahedral (signed permutations) ...
1 vote
61 views

### Show that a permutation equation has 5 solutions

How can I show that this permutation equation has 5 solutions: $\pi^{2013}$ = (1 9) (2 8) (3 7) (4 6) (5) Since the cycle structure is [2, 2, 2, 2, 1] then the only possible cyclic structure for the ...
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### A permutation $\sigma$ of $\{1,2,\dots,14\}$ has $2$ cycles of length $4$ and $2$ cycles of length $3$. How many $\pi$ are there s.t. $\pi^2=\sigma$?

A permutation $\sigma$ of $\{1,2,\dots,14\}$ has $2$ cycles of length $4$ and $2$ cycles of length $3$. How many permutations $\pi$ are there st $\pi^2=\sigma$? So $\pi$ could be made of $1$ cycle of ...
99 views

### If $\binom{n}{2}$ is even, then can you always express the identity permutation as a product of all distinct transpositions in $S_{n}$?

For which $n\geq 2$, is it possible to express the identity permutation as the product of all $\binom{n}{2}$ distinct transpositions in $S_n$? Clearly, we require $\binom{n}{2}$ to be even, which ...
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• 161
1 vote
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### Orders of products of $(n-1)$-cycles in the symmetric group $S_n$

I am interested in the orders of products of $(n-1)$-cycles in the symmetric group $S_n$. In particular what orders of elements can occur as products of $m$ $(n-1)$-cycles for $m=2, 3, 4,\ldots$? It ...
55 views

### What's the inductive hypothesis here?

This question pertains to the proof mentioned in this question. What is the inductive hypothesis here? As I can decipher it, it seems like: If $i$ is the number of terms in 2-cycle identity ...
29 views

### Hamilton paths skipping some vertex relations

I have been developing a Python code to compute all the Hamilton cycles for a system but excluding those that have a maximum distance d between vertices. Hence, for d=1 i would only have a single ...
119 views

### Why are there elements of order $6$ in the permutation group $S_5$? [closed]

I understand that an element in $S_5$ can have an order $6$ if it is product of two disjoint cycles of one of length $2$ and another of length $3$, but I do not understand why these elements have an ...
80 views

### Finding a permutation $a$ such that $a^{-1}(12)(34)a=(56)(13)$

The question is to find the permutation $a$ such that $a^{-1}xa=y$ where $x=(12)(34)$ and $y=(56)(13)$ I found some answers to this question in this site but those don't clarify my doubt. This is my ...
35 views