Abstract

We introduce the notion of a scheduling problem which is a boolean function S over atomic formulas of the form xi ≤ xj. Considering the xi as jobs to be performed, an integer assignment satisfying S schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting function counts the number of solutions to S. We prove that this counting function is a polynomial in the number of time slots allowed. Scheduling polynomials include the chromatic polynomial of a graph, the zeta polynomial of a lattice, the Billera-Jia-Reiner poly-nomial of a matroid and the newly defined arboricity polynomial of a matroid. To any scheduling problem, we associate not only a counting function for solu-tions, but also a quasisymmetric function and a quasisymmetric function in non-commuting variables. These scheduling functions include the chromatic symmetric functions of Sagan, Gebhard, and Stanley, and a close variant of Ehrenborg’s qua-sisymmetric function for posets. Geometrically, we consider the space of all solutions to a given scheduling prob-lem. We extend a result of Steingŕımmson by proving that the h-vector of the space of solutions is given by a shift of the scheduling polynomial. Furthermore, under cer-tain niceness conditions on the defining boolean function, we prove partitionability of the space of solutions and positivity of fundamental expansions of the scheduling quasisymmetric functions and of the h-vector of the scheduling polynomial.