The Combinatorics On Set Partitions Of Type B

Set partitions are classical and wildly studied objects in combinatorics. The the-ory of set partitions mainly contains the investigation of a variety of combinatorial structures of the partitions of a set. In the past, people concern set partitions with specific combinatorial properties, and consider their enumerations, the partition lattice structure, the distributions of some statistics, and so on.In 1995, Reiner first studied set partitions of type Bn. His work mainly con-cerned type B analogue of the lattice of noncrossing set partitions. However, the studies on the set of Bn-partitions itself, including the enumerations of some Bn-partitions with specific properties, the limiting distribution of the number of block pairs, the minimally intersecting property, seem to have received little attention. This thesis completes these gaps, and extends some beautiful prop-erties from ordinary partitions to Bn-partitions. The difficulties of these studies lie mainly on the intensive comprehension of the combinatorial structure of Bn-partitions, and creatively defining reasonable B-analogues of kinds of statistics for Bn-partitions.This thesis is organized as follows. In Chapter 1, we introduce the devel-opment of the theory of set partitions, and briefly recall recent progress on Bn-partitions. In Chapter 2, we give basic enumerative and combinatorial properties of Bn-partitions. In particular, we present a variety of expressions of the num-ber of Bn-partitions without zero-block. As will be seen, this number plays an important role in the study of the combinatorics of Bn-partitions.In the following two chapters, we consider asymptotic properties of the num-ber of block pairs in Bn-partitions. In Chapter 3, we give the exact and asymptotic formulas for the expectation and the variance of the number of block pairs in Bn-partitions. We consider the same problems for Bn-partitions without zero-block. The formulas are new. For the asymptotic formulas, we used complex analysis. In Chapter 4, we establish the normality of the limiting distribution of the num- ber of block pairs in Bn-partitions, as well as in those without zero-block. These results are new. Our main tool is a criterion that requires the infinity of the limit of the variance, which has been established in Chapter 3, and the property that the associated polynomial has only real roots, which can be verified with the aid of the recurrences obtained in Chapter 2.In Chapter 5, we consider minimally intersecting set partitions. We give a formula for the number of minimally intersecting r-tuples of Bn-partitions, as well as of those without zero-block. Our formulas are new. One may regard them as type B analogues of Pittel's results for ordinary set partitions. However, the combinatorial configurations for Bn-partitions are more complicated than those for ordinary partitions. Consequently, we obtain a new proof for Benoumhani's formula, in analogy to Dobinski's formula.In Chapter 6, we demonstrate the symmetric distribution of singletons and adjacencies over set partitions, for which Callan has given a combinatorial proof. Motivated by exploring the equally beautiful property over Bn-partitions, we reasonably define singleton pairs and adjacency pairs for Bn-partitions, and show their symmetric distribution over Bn-partitions without zero-block. In particular, we get a type B analogue of Bernhart's theorem.This thesis contains two appendices. In Appendix A we present some analysis lemmas which are needed in Chapter 2 and Chapter 3. In Appendix B we give a brief introduction to Hayman's theorem, using which one may quickly obtain asymptotic formulas that is weaker than what we get in Chapter 2.