Poset Definitions
'Poset' will often mean 'isomorphism class of posets', a notion which is sometimes indicated by the term 'unlabeled poset'.
Suppose that P is a poset with n elements.
We say P is *labeled* if its elements are named 1,2,..,n.
We say P is *naturally labeled* if whenever x < y in P, then the name of x is less than the name of y in the ordering of the positive integers.
A poset P is *connected* if its Hasse (order) diagram is connected as a graph.
Once you have found Bob Proctor's home page by searching for "Mathematician Robert A. Proctor's Home Page", here are the file names for three definitions and a theorem:
Description of hook length poset: Hook.html
Definition of d-complete poset: DfndC.html
Definition of jeu de taquin poset: JDT.html
Theorems 1 & 2: d-complete implies both jeu de taquin and hook length: index.html
Poset Counts
The number of posets in each class for n=1, 2, 3, 4, 5, 6, 7, ... is:
(Each class except the first class consists of unlabeled posets.)
Naturally labeled posets: 1, 2, 7, 40, 357, 4824, 96428, ...
(Unlabeled) Posets: 1, 2, 5, 16, 63, 318, 2045, 16999, 183231, ...
Connected posets: 1, 1, 3, 10, 44, 238, 1650, 14512, 163341, ...
Posets with unique maximal elements: 1, 1, 2, 5, 16, 63, 318, 2045, 16999, ...
Hook length posets: 1, 2, 4, 10, 23, 63, 165, ...
Connected hook length posets: 1, 1, 2, 5, 11, 31, 75, 232, 607, ...
Indecomposable disconnected hook length posets: 0, 0, 0, 0, 0, 0, 4, 5, 31, ...
Connected d-complete posets: 1, 1, 2, 5, 11, 28, 69, 181, 474, ...
Connected jeu de taquin posets: 1, 1, 2, 5, 11, 32, 77, 236, 616, ...