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.
Each of the following four links exits this website and enters Bob Proctor's complicated website, so use your "Back" button immediately after reading each definition to return to this separate Poset Atlas site!
Definition of hook length poset
Definition of d-complete poset
Definition of jeu de taquin poset
Theorems 1 & 2: d-complete implies both jeu de taquin and hook length
The following link stays within this website:
Definition of indecomposable disconnected hook length poset
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, ...