CHAPTER 11 - A q-Analog of the Partition Lattice

Abstract

This chapter discusses the q-analog of the partition lattice. The set of all partitions of a finite set, when ordered by refinement, is a well-known geometric lattice enjoying a number of structural properties. Every upper interval of a partition lattice is a partition lattice, and every interval is a direct product of partition lattices. The partitions with a single non-trivial block form a Boolean sublattice of modular elements and the Whitney numbers are the familiar Stirling numbers. Because of these and other structural properties, the partition lattices occupy a middle ground between the highly structured modular geometric lattices and arbitrary geometric lattices, exhibiting some of the consequences of the departure from modularity while still retaining enough of the structure to facilitate their study. The chapter describes a prime power q a class of geometric lattices, called q-partition lattices, which share a number of the properties of partition lattices.