m-Level rook placements

TL;DR

This paper generalizes rook placement theorems to m-level Ferrers boards, introduces q,p-analogues, and explores properties of equivalence classes, connecting to q,t-Catalan numbers and open questions.

Contribution

It extends factorization theorems for rook placements to all Ferrers boards and introduces weighted file placements and analogues, broadening the theoretical framework.

Findings

Generalized rook placement theorems to all Ferrers boards.

Introduced q,p-analogues and analyzed equivalence classes.

Connected rook placements to q,t-Catalan numbers.

Abstract

Goldman, Joichi, and White proved a beautiful theorem showing that the falling factorial generating function for the rook numbers of a Ferrers board factors over the integers. Briggs and Remmel studied an analogue of rook placements where rows are replaced by sets of $m$ rows called levels. They proved a version of the factorization theorem in that setting, but only for certain Ferrers boards. We generalize this result to any Ferrers board as well as giving a p,q-analogue. We also consider a dual situation involving weighted file placements which permit more than one rook in the same row. In both settings, we discuss properties of the resulting equivalence classes such as the number of elements in a class. In addition, we prove analogues of a theorem of Foata and Sch\"utzenberger giving a distinguished representative in each class as well as make connections with the q,t-Catalan numbers. We end with some open questions raised by this work.