In this thesis, we use permutation tableaux of type B to enumerate known statistics of signed permutations and to give two combinatorial proofs.
We realize many important statistics of signed permutations in the corresponding permutation tableaux or bare tableaux of type B: such as alignments, crossings, inversions, and cycles. This enables us to describe how to relate the number of alignments and crossings with other statistics of signed permutations and also to illustrate the covering relation in weak (Bruhat) order on Coxeter system of type B in terms of permutation tableaux of type B.
We give two combinatorial proofs in the form of algorithms. One is a proof of a symmetry of (t,q)-Eulerian numbers of type B. We dene an involution preserving many important statistics on the set of permutation tableaux of type B. This also proves a symmetry of the generating polynomial of the numbers of crossings and alignments, and hence q-Eulerian numbers of type A defined by Williams. By considering a restriction of our bijection, we were led to dene a new statistic on the permutations of type D and (t,q)-Eulerian numbers of type D, which is proved to have a particular symmetry as well. We conjecture that our new statistic is in the family of Eulerian statistics for the permutations of type D. The other is a proof of equidistribution of alternating signed permutations with the maximal number of fixed points and derangements. A conjecture of Stanley on a class of alternating permutations, which is proved by Chapman and Williams, states that alternating permutations with the maximal number of fixed points is equidistributed with derangements. We extend this (type A) result to type B we prove that various classes of alternating signed permutations with the maximal number of fixed points are equidistributed with certain types of derangements of type B respectively.
As an additional topic, we study the cohomological rigidity of n-simple polytopes with n+3 facets obtained from Gale-diagram on a pentagon and find two such polytopes which show that cohomological rigidity does not imply algebraic rigidity.
