"Rook and Wilf equivalence of integer partitions"

Abstract:  The subjects of rook equivalence and Wilf equivalence have both attracted considerable attention over the last half-century.  We will start with a review of Rook Theory, i.e., the study of the number of configurations of k-rooks on a broken chess board, as established by Kaplansky and Riordon in the 1940's and later characterized by Foata and Shutzenberger in the 1970's.  We then introduce a new notion of Wilf equivalence for integer partitions and also discuss a characterization of the related generating function.  This alternate characterization allows us to prove that rook equivalence implies Wilf equivalence. We are also able to prove that if we refine the notions of rook and Wilf equivalence in a natural way, then these two relations coincide.