I was trying to derive an expression to solve the following problem: "How many ways are there to arrange $n$ non-attacking kings on an $n X n$ board, with $1$ in each row and column."
There is an OEIS sequence for this problem : https://oeis.org/A002464
Using inclusion-exclusion, I was able to derive the following expression:
Define $\displaystyle \sigma_n(k) = n-k$ for $n < 2k$ and $\displaystyle \sigma_n(k) = k$ for $n \geq 2k$
Then $\displaystyle u_n = n! - \sum_{k=1}^{n-1}(\sum_{i=1}^{\sigma_n(k)} {k-1 \choose i-1}{n-k \choose i}(2^i))(n-k)!(-1)^{k-1}$
I was able to compute this for $n=4$, and got the correct answer of $2$, however my $\sigma_{n}(k)$ term makes it difficult for me to compute this for larger $n$, and I'm not sure how to get rid of it, and in general overcome this obstacle.
I am also interested in what this expression is asymptotic to as $n \to \infty$ but again find it hard to work this out with my $\sigma_{n}(k)$ term
Any and all help would be much appreciated!
EDIT 1: After checking for $n=5$ I can confirm my expression for $u_n$ also produces the correct answer for this case