On the orthogonal product of simplices and direct products of truncated Boolean lattices

Abstract

The initial point of this paper are two Kruskal–Katona type theorems. The first is the Colored Kruskal–Katona Theorem which can be stated as follows: Direct products of the form B k 1 1× B k 2 1×⋯× B k n 1 belong to the class of Macaulay posets, where B k t denotes the poset consisting of the t+1 lowest levels of the Boolean lattice B k . The second one is a recent result saying that also the products B k 1 k 1−1 × B k 2 k 2−1 ×⋯× B k n k n −1 are Macaulay posets. The main result of this paper is that the natural common generalization to products of truncated Boolean lattices does not hold, i.e. that ( B k t ) n is a Macaulay poset only if t∈{0,1, k−1, k}.