I am trying to find an expression for the number of permutations on $n$ elements with no fixed points. I know that this can be done using the inclusion-exclusion principle, but I am trying to do this another way.
First I derived a recurrence $F_n$ for the number of permutations on n elements. If we choose to send the number 1 to a specific number j, we can partition the collection of permutations that do this into two sets: (1) those that send j back to 1 and (2) those that don’t.
In (1) we’ve mapped 1 to j and j to 1, so what remains is to assign the numbers in {2, …, j-1, j+1,…, n} to numbers {2, …, j-1, j+1,…, n}, which is the number of permutations with no fixed points on n-2 elements, or $F_{n-2}$.
In (2), we’ve mapped 1 to j, but need to map the remaining numbers {2, …, n} to the numbers {1, …, j-1, j+1, n} in such a way that j does not map to 1; we can relabel 1 as j’, and now this reduces to permutations with no fixed points on n-1 elements, or $F_{n-1}$.
Since we have $(n-1)$ choices of j, this results in the recurrence $$F_n = (n-1)(F_{n-1} + F_{n-2})$$
To solve this recurrence I tried to use the method of generating functions, writing,
$F = \sum_{k=0}^{\infty} F_{k}x^k = \sum_{k=0}^{\infty} (k-1)(F_{k-1} + F_{k-2})x^k =$
$\sum_{k=0}^{\infty} (k-1)F_{k-1}x^k + \sum_{k=0}^{\infty} (k-1)F_{k-2}x^k$
Then writing,
$x^2F’ = \sum_{k=0}^{\infty} kF_{k}x^{k+1} = \sum_{k=1 \to \infty} (k-1)F_{k-1}x^{k}$
$x^2(xF)’ = \sum_{k=0 \to \infty} (k+1)F_{k}x^{k+2} = \sum_{k=2 \to \infty} (k)F_{k-2}x^{k}$
Then,
$F = x^2F’ + x^2(xF)’ = x^2F’ + x^2[F + xF’] = x^2F’ + x^2F + x^3F’$
And so we have the differential equation, $$F = [x^2/(1-x)]F’$$
I tried to solve this differential equation, and hopefully be able to find the power series expansion for the solution, so that I can equate its coefficients with $F_{k}$.
This is where things didn’t really work out — defining $F_{0} = 1$, I have the initial condition $F(0) = 1$. Wolfram Alpha gives the solution to this differential equation as:
$$\frac{C_1e^{-1/x}}{x}$$ which is not defined at 0.
If I could get a hint on how to proceed here I would appreciate it!
