The Radon transforms of the combinatorial geometry. II: Partition lattices
[For Part I see Zbl 0406.05023.] The paper studies lattices of partitions. The lattice of partitions of the number set \(\{1,\dots,n\}\) is denoted by \(\Pi_ n\). The rank of \(\pi \in \Pi_ n\) is equal to \(n\) minus the number of classes of \(\pi\). The incidence matrix \({\mathcal M} (\Pi_{n+1};\;k,k+1)\) is considered. The columns (or rows) of this matrix correspond to elements of \(\Pi_{n+r}\) of rank \(k\) (or \(k+1\) respectively). An entry of the matrix is 1, if the partition corresponding to the row is greater than that corresponding to the column in the lattice \(\Pi_{n+1}\); otherwise it is 0. The main theorem states that for \(k<n/2\) the matrix \({\mathcal M} (\Pi_{n+1};k,k+1)\) has rank \(W(\Pi_{n+1};k)\), i.e. the number of elements of rank \(k\) in \(\Pi_{r+1}\) (called Whitney number of the second kind).
scientific article; zbMATH DE number 495499