Draw figures for the 5 different lattices with 5 elements.

Question

I can only think of 4 lattices. Those are: a 1-1-1-1-1 (a chain), a 1-3-1, a 1-1-2-1 and a 1-2-1-1 (if this notation isn't clear, I'll provide images). I really can't figure out what the 5th lattice would be, thank you.

Answer 1

You seem to be missing the lattice where you have $a<c<e$ and $a<b<d<e$ but $c$ is incomparable to $b$ and $d$. I.e. there are two paths from the bottom to the top but the one is of pathlength two and the other is of pathlength three.