Statistics On Finite Coxeter Groups

Permutations are among the richest objects in combinatorics. We can bijectively associate them with other structures, such as trees, lattice paths, noncrossing partitions,01-matrices, standard Young tableaux and so on. The modern study of permutation statistics began with the work of P.A. MacMahon. There has been much work done in studying distributions of permutation statistics, such as maj, inv, des, exc, etc. It is known that the symmetric group of permutations is a Coxeter group of type A. Many properties of permutation statistics have been extended to the Coxeter groups of type B and type D.The main objective of this thesis is to research the distributions of both integer-valued statistics and set-valued statistics on Coxeter groups of types A, B and D. Our contribution is briefly summarized as follows. Firstly, we find a combinatorial interpre-tation of an equidistribution result on permutations by employing a bijection of D. Foata and G.-N. Han on permutations. This answers a question of T.K. Petersen. We also give an explicit combinatorial interpretation of an equidistribution result of S. Poznanovic on restricted permutations. We further extend some permutation statistics to standard Fibonacci tableaux. Secondly, we construct a bijection on Bn and obtain several dis-tribution results concerning some new set-valued statistics. This gives a generalization of several distribution results of Foata and Han on permutations. We also derive some refinements of these conclusions by considering a decomposition and some restrictions on signed permutations. Lastly, we introduce two new statistics on even-signed permu-tations and deduce a strengthened form of an equidistribution result of Petersen. This thesis consists of four chapters.In Chapter1, we give a review of the background of this work as well as some ba-sic knowledge. To be specific, we introduce the Coxeter system, the symmetric group, Dyck path, perfect matching and so on. Meanwhile, some well-known generating func-tions are exhibited.In Chapter2, we focus on some distribution properties of permutation statistics. In the second section, we use a bijection of Foata and Han to interpret an equidistribution of Petersen combinatorially. In the third section, we give a proof of an equidistribution result on the set of permutations that correspond to arrangements of n non-attacking rooks on a Ferrers board with n rows and n columns. In the last section, we will extend some permutation statistics to standard Fibonacci tableaux.In Chapter3, we introduce several set-valued statistics for signed permutations of Bn and obtain many distribution results corresponding to them. We construct a bijection on Bn to show that the six quaternaries of set-valued statistics (CycB, RmilB, CycB, RmilB),(CycB, LmapB, CycB, LmapB),(RmilB, LmapB, RmilB, LmapB),(LmapB, RmilB, LmapB, RmilB),(LmapB, CycB, LmapB, CycB) and (RmilB, CycB, RmilB, CycB) are equidistributed over Bn. These results serve as type B analogues of equidistribution results of Foata and Han on permutations. More-over, we show that these equidistributions can be refined by virtue of a decomposition of signed permutations. At the end of this chapter, we also consider the equidistribution property of two quintuples of statistics on restricted signed permutations.In Chapter4, we introduce two new statistics on even-signed permutations, and obtain an equidistribution result of two pairs of statistics by constructing a bijection on Dn. This gives a refinement of an equidistribution result of Petersen. We also give an algebraic proof of this result. To achieve this goal, we apply two factorizations of the digonal sum of Dn and obtain explicit formulas for the bivariate generating functions of these two pairs of statistics.