# Shellable simplicial complex with restriction on shellings

Does there exist a (pure) shellable simplicial complex $\Delta$ with the following property? There is some facet $F$ of $\Delta$ such that no shelling can begin with $F$.

This condition is easily seen to be equivalent to the existence of a shellable simplicial complex $\Delta'$ with at least two facets, such that for some facet $F'$ every shelling of $\Delta'$ ends with $F'$.

• If such a facet $F$ exists, then is it true that $F$ contains a shedding vertex? – T. Amdeberhan Dec 21 '16 at 21:23
• Does Example 1.17 work from math.cornell.edu/~eranevo/homepage/FaceRingNotes.pdf – T. Amdeberhan Dec 21 '16 at 21:29
• @T.Amdeberhan Example 1.17 does not work. In fact, it is known that the smallest $f$-vector of a shellable simplicial complex for which some partial shelling cannot be extended to a shelling is $(6,14,9)$. – Richard Stanley Dec 22 '16 at 3:25

The $$f$$-vector of this complex is (1,7,19,13). I suspect that 7 vertices is minimal.