A note about shellable planar posets

Duško Jojić

We will show that shellability, Cohen-Macaulayness and vertex-de composability of a graded, planar poset $P$ are all equivalent with the fact that $P$ has the maximal possible number of edges. Also, for a such poset we will find an $R$-labelling with $\{1,2\}$ as the set of labels. Using this, we will obtain all essential linear inequalities for the flag $h$-vectors of shellable planar posets from [1].