1

i have been asked to solve the quastion , when i tried and i found it better to prove first the lemma (both the lemma and the quastion need to be proven):

let $\sigma \in S_n$ be a permutation from {1,...,n} to {1,...,n} so prove that,

lemma: $\forall x \in ${1,...,n}$ : \sigma (x) \neq x \ so \ \sigma (\sigma(...(\sigma(x))...) [n \ times] = x \ \ , \forall x \in${1,...,n}

now the question is : whats the probability that by choosing arbitrary permutation $\sigma (\sigma(...(\sigma(x))...) [n \ times] = x \ \ , \forall x \in${1,...,n} ?

please try to prove them both but even a solution to the lemma alone will be excepted - thanks for help.

ned grekerzberg
  • 977
  • 3
    The arrow $\Leftarrow$ in the lemma is wrong. Choose $\sigma$ to be the identity, then $\sigma^n(x)=x$ obviously, but also $\sigma(x)=x$. – Luke Oct 12 '17 at 09:20
  • thanks, yet it doesn't metter for the outcome – ned grekerzberg Oct 12 '17 at 09:59
  • For the probability part supposing I have understood it correctly there is this MSE link. In fact we don't need inclusion-exclusion from the link as your question appears to count permutations whose order divides $n$ rather than being exactly $n.$ – Marko Riedel Oct 12 '17 at 20:46

1 Answers1

1

No answer but too much for a comment.

A route to solve the question is the following: find the tuples $\langle a_1,\dots,a_n\rangle$ of nonnegative integers exist with:

  • $\sum_{k=1}^n a_k\cdot k=n$
  • $\mathsf{lcm}(a_1,\dots,a_n)\mid n$

Here $a_k$ denotes the number of cycles of length $k$ in the complete disjoint cycle presentation of $\sigma$.

A tuple $\langle a_1,\dots,a_n\rangle$ that satisfies the conditions stands for a conjugacy class that contains elements $\sigma$ that satisfy $\sigma^n=\mathsf{id}$ and the number of elements of the class is: $$\frac{n!}{\prod_{k=1}^nk^{a_k}\cdot a_k!}$$

So if $A$ denotes the collection of these tuples then we find a probability: $$\sum_{\langle a_1,\dots,a_n\rangle\in A}\frac{1}{\prod_{k=1}^nk^{a_k}\cdot a_k!}$$

I hope that there is simpler route.

drhab
  • 151,093