If t is a positive integer, then a partition of a non-negative integer n is a t−core if none of the hook numbers of the associated Ferrers-Young diagram is a multiple of t. These partitions arise in the representation theory of finite groups and also in the theory of class numbers. We prove that if t = 2, 3, or 4, then two different t−cores are rook equivalent if and only if they are conjugates...

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook configurations to give a combinatorial interpretation to the q-analogue of a formula of Frobenius relating the Stirling numbers of the second kind to the Eulerian polynomials. Later, Remmel and Wachs defined generalized p, q-Stirling numbers of the first and second kind in terms of rook placements. Additionally, they extended their definit...

First, we look at the distribution of permutation statistics in the context of pattern-avoiding permutations. The first part of this chapter deals with a recursively defined bijection of Robertson [6] between 123and 132-avoiding permutations. We introduce the general notion of permutation templates and pivots in order to give a non-recursive pictorial reformulation of Robertson’s bijection. Thi...

We develop two q-analogues of the previously defined poly-Stirling numbers of the first and second kinds. We also develop the corresponding q-rook theory models to give combinatorial interpretations to these numbers.