Konrad Engel, Uwe Leck
Optimal antichains and ideals in Macaulay posets
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock

MSC:

06A07 Combinatorics of partially ordered sets

Abstract: A ranked poset P is a Macaulay poset if there exists a linear ordering
<= of its elements such that for any i and any subset F of the i-th
level N_i the shadow of the |F| smallest elements of N_i w.r.t. <=
is contained in the set of the smallest |\delta(F)| elements of N_i-1,
where \delta(F) denotes the shadow of F. We consider the following
three optimization problems for P: (i) Find the maximum weight of an
ideal of given size. (ii) Find the minimum weight of an ideal generated
by an antichain of given size. (iii) Find the maximum number of maximal
elements of an ideal of given size. We provide solutions for a class
of Macaulay posets which includes all examples known so far.
Keywords: Macaulay posets, shadow, antichain, ideal