Here is a counting problem based on Gilbreath permutations.
You have 10 penguins, each a different height. You want to take a photo of the penguins in a single straight line. First though, you select some number of the penguins to be looking slightly to the right, and the others to be pointing slightly to the left. For fun, you decide that all the right-looking penguins should be increasing in height while all the left-looking penguins should be decreasing in height, as you move from right to left. How many different such arrangements are possible?