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ON THE FLAG f-VECTOR OF A GRADED LATTICE WITH NONTRIVIAL HOMOLOGY
 

Summary: ON THE FLAG f-VECTOR OF A GRADED LATTICE WITH
NONTRIVIAL HOMOLOGY
CHRISTOS A. ATHANASIADIS
Abstract. It is proved that the Boolean algebra of rank n minimizes the flag f-vector
among all graded lattices of rank n, whose proper part has nontrivial top-dimensional
homology. The analogous statement for the flag h-vector 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
f-vector; 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 cd-index, an invariant which refines the flag
f-vector, 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

  

Source: Athanasiadis, Christos - Department of Mathematics, University of Athens

 

Collections: Mathematics