What can be said about pure O-sequences?

Abstract

Let f( Δ) = ( f 0, f 1, …, f d−1 ) be the f-vector of a Cohen-Macaulay complex Δ. Björner proved that ( ∗) f i ⩽ f ( d−2)− i for any 0⩽i<[ d 2 ] and ( ∗∗) f 0⩽f 1⩽ … ⩽f [ (d−1) 2 ] . Recently, Stanley generalized Björner's inequalities ( ∗) and ( ∗∗) for pure simplicial complexes. In this paper we consider O-sequence analogue of the inequalities ( ∗) and ( ∗∗). Let ( h 0, h 1, …, h s ), h s ≠0, is a pure O-sequence. We shall prove that h i ⩽ h s− i for any 0⩽i⩽[ s 2 ] and h 0⩽h 1⩽ … ⩽h [ (s+1) 2 ] .