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Scheduling Problems

We introduce the notion of a scheduling problem which is a boolean function $S$ over atomic formulas of the form $x_i \leq x_j$. Considering the $x_i$ 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 polynomial of a matroid. To any scheduling problem, we associate not only a counting function for solutions, 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 quasisymmetric function for posets. Geometrically, we consider the space of all solutions to a given scheduling problem. We extend a result of Steingrímmson by proving that the $h$-vector of the space of solutions is given by a shift of the scheduling polynomial. Furthermore, under certa