Cycle-counting Q-rook theory and other generalizations of classical rook theory

In this dissertation, we will study some generalizations of classical rook theory. The main focus is what we call cycle-counting q-rook theory, a model which incorporates a weighted count of both the number of cycles and the number of inversions of a permutation. In Chapter 1 we discuss background material in rook theory. In Chapter 2, we prove some results about algebraic properties of a generalization of the cycle-counting q-hit numbers. The main result of the dissertation is presented in Chapter 3, a statistic for combinatorially generating the cycle-counting q-hit numbers, which were previously defined only algebraically. We will then apply this result to prove some new theorems about permutation statistics involving cycle-counting in Chapter 4. Finally, in Chapter 5 we explore two other models which generalize classical rook theory and prove some basic results about these models. The first is a refinement of cycle-counting q-rook theory, obtained by adding a new parameter p to the weight of each rook placement. The second is a theory in which more than one rook is allowed in a row, but a row containing more than one rook is weighted by a polynomial in α