Boolean Antichains in Finite Lattices: Product Formulas and Matroid Applications

We study the enumeration of Boolean antichains in finite lattices through product formulas and matroid correspondences. The binomial transform of the labelled Boolean-antichain counts is multiplicative under lattice products, yielding an explicit positive convolution for every size. For a rank-r matroid, maximum Boolean antichains in its flat lattice are naturally in bijection with bases of the simplification. This inverse correspondence extends the known Boolean span of a basis and proves the spanning-tree expectation of Garber, Goltermann, Horiatakis, König, and Gottesman in the connected simple graphic case.

The framework gives explicit all-size counts for uniform matroids and finite subspace lattices. Parallel-class weights recover the multivariate basis polynomial, and flat intervals give minor-local and rank-tight versions. For an arbitrary finite lattice, an exact recognition criterion tests whether the candidate Boolean atoms rise above the common meet and all 2^k reconstruction height gaps vanish. Summing these tests over ordered tuples gives the complete enumerator, while the structured families above admit more compact evaluations. Further consequences include a disjoint-filter interpretation for distributive lattices, a universal Möbius formula for Boolean pairs, and complete distributions for finite projective planes.