Characterization of magic positivity via capacity and demand

Determine a complete characterization of when the Snapper polynomial \(\chi_{M,P}(q)\) associated with a lattice generalized permutohedron \(P\) on a loopless matroid \(M\) is magic positive, expressed in terms of the capacities and demands of the supported flats.

Background

For a lattice generalized permutohedron $P$ on a loopless matroid $M$, the paper associates nonnegative weights $\omega_{M,P}(F)$ to supported flats $F$. Each supported flat has a capacity $\operatorname{cap}_F=\operatorname{rk}_M(F)-1$ and a demand $\operatorname{dem}_F$, defined from the maximal dragon Hall--Rado tuples supported on the weight support. These quantities define sufficient and necessary conditions for magic positivity: saturation, requiring $\omega_{M,P}(F)\geq \operatorname{cap}_F$, implies magic positivity, while magic positivity implies weak saturation, requiring $\omega_{M,P}(F)\geq \operatorname{dem}_F$.

The examples in the paper show that the implications between zonotopal, saturated, magic positive, and weakly saturated classes are strict. Thus, the existing capacity- and demand-based criteria do not coincide in general, and the authors leave unresolved the problem of finding a complete criterion for magic positivity in this framework.