Saturated chain partitions in ranked partially ordered sets, and non-monotone symmetric 11-Venn diagrams

In this paper we show that there are at least 2110 non-isomorphic 11-doilies, that is, there are many non-isomorphic symmetric, non-simple, non-monotone 11-Venn diagrams, with "many" vertices. We do not achieve the maximum vertex set size, 2046, but we approach it closely, improving from the previous 462 in [10] to 1837. The doilies constructed here cannot be constructed by either of the methods of [10] or [6]. The main purpose of this paper is not to publish these attractive diagrams but to inspire new studies by raising ideas, methods, questions, and conjectures, hoping for results analogous to those generated in [10]. These ideas connect two seemingly distant areas of mathematics: a special area of combinatorial geometry, namely, certain families of simple closed Jordan curves in the plane, and the study of ranked partially ordered sets or posets.