I have a flat cone in which I can drop marbles. There is space for one marble in the bottom row, two marbles in the row above that, three marbles above that, and so on.
\ / \5 / \3 6 / \2 4/ \1/
A marble with the number $x$ means it was the $x$th marble to be dropped. The marbles follow the laws of gravity, which means that a given position can be occupied by an incoming marble only if another marble (or a wall of the cone) exists to both its bottom left and bottom right.
The marbles can pile over the top of the cone like this:
8 \ 7 4/ \ 6 3/ \5 2/ \1/
but, in this case, the maximum number of marbles you can put is $16 = 4 \times 4$. In general, you are allowed to put $ab$ marbles where $a$ and $b$ are the lengths of the sides of the cone.
My question is, if the sides of the cone are of length $a$ and $b$, then in how many ways can you drop all the $ab$ marbles into the cone? This could possibly be solved more specifically for $a = b$, or more generally for three-dimensional cones.
(This question is inspired by this one, which is (in my opinion) the special case $a = b = 3$.)
Edit: Since the flat cone case is in easy bijection to standard Young tableaux as pointed out by @Marcel in the answer below, can anybody suggest some approaches for three-dimensional $k$-gonal pyramids? Just tetrahedrons, maybe?