 
Summary: ON THE FLAG fVECTOR OF A GRADED LATTICE WITH
NONTRIVIAL HOMOLOGY
CHRISTOS A. ATHANASIADIS
Abstract. It is proved that the Boolean algebra of rank n minimizes the flag fvector
among all graded lattices of rank n, whose proper part has nontrivial topdimensional
homology. The analogous statement for the flag hvector is conjectured in the Cohen
Macaulay case.
1. Introduction
Let P be a finite graded poset of rank n 1, having a minimum element ^0, maximum
element ^1 and rank function : P N (we refer to [12, Chapter 3] for any undefined
terminology on partially ordered sets). Given S [n  1] := {1, 2, . . . , n  1}, the number
of chains C P {^0, ^1} such that {(x) : x C} = S will be denoted by fP (S). For
instance, fP (S) is equal to the number of elements of P of rank k, if S = {k} [n  1],
and to the number of maximal chains of P, if S = [n  1]. The function which maps S to
fP (S) for every S [n  1] is an important enumerative invariant of P, known as the flag
fvector; see, for instance, [4].
The present note is partly motivated by the results of [2, 6]. There it is proven that
the Boolean algebra of rank n minimizes the cdindex, an invariant which refines the flag
fvector, among all face lattices of convex polytopes and, more generally, Gorenstein*
lattices, of rank n. It is natural to consider lattices which are not necessarily Eulerian, in
