de Mathématiques de Luminy
Ille Pierre, Rampon Jean-Xavier.
A counting of the minimal realizations of the posets of dimension two
The posets of dimension 2 are those posets whose minimal realizations have two elements, that is, which may be obtained as the intersection of two of their linear extensions. Gallai's decomposition of a poset allows for a simple formula to count the number of the distinct minimal realizations of the posets of dimension 2. As an easy consequence, the characterization of M. El-Zahar and of N.W. Sauer of the posets of dimension 2, with an unique minimal realization, is obtained.
Key words : Counting ; Dimension ; Directed Graphs ; Gallai's Partition ; Indecomposable ; Posets ; Realization.