The number of tiered posets modulo six

Our goal in this note is to sketch a proof of a conjecture made by Kreweras [1] in a paper published in this journal. A poset on an n-set is said to be tiered with height h if every element belongs to a maximal chain with exactly h elements. It is easy to visualize these posets in terms of their Hasse diagrams. The elements can be arranged in tiers with the minimal elements in the first tier and the maximal elements in the top tier. In a pair of consecutive tiers every element in the upper tier covers some element in the lower tier, and every element in the lower tier is covered by some element in the upper tier. A typical tiered poset is shown in Fig. 1. A pair of consecutive tiers of a tiered poset forms a bipartite graph in which every vertex has positive degree. We showed in [2] that f ( m , n) , the number of bipartite graphs having m lower vertices and n upper vertices, all vertices with positive degree, is given by