Four-element Generating Sets with Block Count Widths at Most Two in Partition Lattices

The partitions of a finite set form a so-called partition lattice. Henrik Strietz proved that this lattice has a four-element generating set; his paper has been followed by a dozen others. Two recent papers of the present author indicate that small generating sets of these lattices can be applied in cryptography. The block count of a partition is the number of its blocks. Given a four-element set of partitions, list the block counts of its members in increasing order. Then subtract the first (i.e., the smallest) block count from all four to obtain the components of a four-dimensional vector. This vector and its last component are called the block count type and the block count width, respectively, of the given four-element set in question. There are exactly ten block count types of width at most two. We prove that for any partition lattice over a finite base set with at least eight elements, each of the ten block count types of width at most two is the block count type of a four-element generating set of the partition lattice; moreover, we give a lower bound of the number of these generating sets.