What does $!n$ mean where $n$ is any positive integer?

Question

I know what a factorial is, but I am taking combinatorics now and one of the solutions to a problem contained something like this $!6$, this is the first time I have seen something like this.

What does this mean and how do I evaluate it?

Answer 1

I don't think it's universally accepted notation, but it usually stands for a derangement number. Thus $!n$ denotes the number of permutations of $\{1,\cdots, n\}$ with no fixed points.

It satisfies the recursion $$!n=(n−1)\left(!(n−1)+!(n−2)\right)$$ ($n!$ satisfies the parallel one but $!0=1,!1=0$ whereas $0!=1=1!$)