I'm trying to solve a couple of problems from Stanley's Enumerative Combinatorics Vol 1, 2nd Edition and feel quite clueless about how to approach them:
Q34. Find all non-isomorphic posets P such that $F (J (P ), x) = (1 + x)(1 + x^2 )(1 + x + x^2 )$
It is clear after expanding the RHS that the coefficients are 123321 for $x^0, x^1, \ldots , x^5$ and the set has 5 elements, but I'm not sure if there is a intuitive or direct way of finding ALL the non isomorphic posets. Any help would be appreciated!
Q46a. Let $f(n)$ be the number of sublattices of rank n of the Boolean algebra $B_n$. Show that $f(n)$ is also the number of partial orders $P$ on $[n]$.
I'm completely stuck on how to go about this proof. As fas I could tell, there is no explicit formula for the number of partial orders on [n]. How could one determine $f(n)$?
PLEASE let me know your suggestions. Thank you so much for all your help!