Thank you for all your comments sincerely! I think my problems are solved.

Thanks for comments. I think the poset $M(n)$ in Stanley's article "Some applications of algebra to combinatorics" is exactly what I described above. However I have a pool knowledge about the rank generating function, is there any way to get the maximal rank size from this function? There are only some conclusions about $M(n)$ in this article, is there any reference books which can help me understand better？