The Relative Derangement And The Set Partition On B_n

The main results of this thesis are to discuss some combinatorial properties about derangements and set partitions on the hyperoctahedral group Bn. The derangements and the set partitions are the two important subjects in Combina-torics. Many scholars in the combinatorial area have generalized their properties from the symmetric group Sn to the hyperoctahedral group Bn. The thesis has two parts. The first part is to investigate the relation between the relative de-rangements and the derangements on Bn, and give a combinatorial interpretation for the relation. The second part is to study set partitions on Bn. We construct a new statistics and give a generating function of this statistics.In Chapter 1, we give an evolution of the background on derangements and set partitions on Bn, and introduce some definitions and notations that are used throughout the thesis.In Chapter 2, we give an overview of some important conclusions on derange-ments and set partitions, including a famous transformation proposed by Foata and Guoniu Han [33] called the fundamental transformation for signed words, the relative derangements and the skew derangements on the symmetric group Sn, and the pmaj index for set partitions on the symmetric group Sn.Chapter 3 is the first part of this thesis. In this chapter, we mainly study the relation between relative derangements and derangements on the hyperocta-hedral group and give a combinatorial proof. We introduce a new derangement on Bn, which is called the skew derangement on Bn or the signed skew derange-ment. By constructing a one-to-one correspondence, we prove that any relative derangement of type B has a corresponding skew derangement of type B. Then, we indicate that any skew derangement of type B can be transformed to a signed derangement. In this way, we complete the proof of the relation between relative derangements and derangements on Bn, by means of this intermediate variable, the signed skew derangements.Chapter 4 is the second part of this thesis. In this chapter, we mainly study the set partitions on the hyperoctahedral group Bn. We construct a new statistics for set partitions on Bn called p-flag major index via defining a group represen-tation of set partitions. And we present a generating function on this statistics. Then, we introduce 2-crossing numbers of type B based on the group representa-tions. And we prove the two statistics are equi-distributed on the set PnB(S, T).