Situation:
We want to build a wall. The lowest level of this wall consists of 9 stones. Every stone ( except the stones of the lowest level) has to be on the middle of two other stones. Moreover we have no gaps regarding the lowest level. Question: how many options do we have for a wall like this?
my idea\work:
I made a picture of this situation.
I think this has to do with the Catalan numbers $C_n$. Maybe "up" is like opening a parantheses and " going right" is like closing a parantheses? I think the solution is $C_9$. Am I right? If I'm Right: Can you please help me to improve my attempt. It is a bit messy and not detailed.
* Second Explanation * Maybe just going up is to open a parantheses and going down again is like Closing a parantheses. But I'm not sure if $C_9$ is correct. I mean in my picture Im going only 6 times up and 6 times down.
It seems correct to me.
If we give the utmost left brick the coördinates $(0,0),(0,1),(2,0),(2,1)$.
Then starting at $(0,0)$ add arrow $\nearrow$ to your picture, going from $(0,0)$ to $(1,1)$.
Under the criterium that the arrows must be pictured on the wall repeat this, and if such arrow is not possible then add arrow $\searrow$ (which can be done on the wall).
Then a path comes in sight starting at $(0,0)$ and ending at $(18,0)$ where the cornerpoints have nonnegative coördinates.
There are $C_9$ of such paths.