| The statistics des; inv; maj are well-known statistics on Sn. A central theme of this dissertation is to extend these statistics and others to compositions. A composition, or word on P , the set of positive integers, is simply a sequence of positive integers gamma 1, gamma2,...gamman. In Chapter 3, we derive generating functions for basic composition statistics such as des, inv, maj, as well as statistics unique to compositions such as lev and levmaj, defined by Levg= i:gi=gi+1 , levg= Levg ,and levmajg= i∈Levg i. A permutation sigma ∈ Sn is called up-down if sigma1 < sigma 2 > sigma3 <···. When we consider analogues for up-down words, we find that there are four classes to consider: strict or weak increase, followed by strict or weak decrease. We derive generating functions for all four classes in Chapter 4, and we also generalize previous results to words that have a weakly/strictly increasing block of length s followed by a weak/strict decrease. In order to handle all classes, we use an involution that reduces the original classes considered to ones that are easier to count. In Chapter 5, we generalize these results by forcing the final letter in each block of length s to be in some set X ⊂ P.;In Chapter 6, we apply an alternate method to find the generating functions for certain classes of alternating words on alphabet P. In addition, we use the results of previous chapters to find generating functions for statistics defined by altdesw =2i:w2 i>w2i+1 ∪ 2i+1:w2+1<w2 i+2 , waltdesw =2i:w2 i≥w2i+1 ∪ 2i+1:w2i+1 ≤w2i+2 , altmajw =i∈Altdesw waltmajw =i∈Waldesw i.;Finally, in Chapter 7 we find generating functions for additional composition patterns. For instance, we can partition a composition into blocks of fixed length and count levels between the maxima of these blocks. |
