arXiv subjects
Publications and source records attributed to Francesco Marigo.
First a few reformulations of Frankl's conjecture are given, in terms of reduced families or matrices, or analogously in terms of lattices. These lead naturally to a stronger conjecture with a neat formulation which might be easier to attack than Frankl's. To this end we prove an inequality which might help in proving the stronger conjecture.
We characterize the finite distributive lattices which admit a complete valuation, that is bijective over a set of consecutive natural numbers, with the additional conditions of completeness (Definition 2.3). We prove that such lattices are downset lattices of finite posets of dimension at most two, and determine a realizer through a recursive relation between weights on the poset associated to valuation. The relation shows that the weights count chains in the complementary poset. Conversely, we prove that a valuation defined on a poset of dimension at most two, through the weight function which counts chains in the complementary poset, is complete.
An open partition π [Cod09a, Cod09b] of a tree T is a partition of the vertices of T with the property that, for each block B of π, the upset of B is a union of blocks of π. This paper deals with the number, NP(n), of open partitions of the tree, V_n, made of two chains with n points each, that share the root.