Large-order distinguishability of selected specializations

Determine whether the specializations \(\Phi(\cdot;1,1,x,y,z)\), \(\Phi(\cdot;s,t,1,1,z)\), \(\Phi(\cdot;1,t,xw,1,1/w)|_{w=0}\), \(\Phi(\cdot;s,1,1,yw,1/w)|_{w=0}\), \(\Phi(\cdot;1/w,1,x,y,w)|_{w=0}\), and their dual variants distinguish series-parallel posets of arbitrarily large cardinality.

Background

Several polynomial specializations distinguish all series-parallel posets with at most ten elements according to the authors’ computations. However, the computations do not establish whether this property persists for larger posets. The unresolved issue concerns the behavior of these specific specializations beyond the tested finite range.

References

The specializations \Phi(\cdot ; 1, 1, x, y, z), \Phi(\cdot ; s, t, 1, 1, z), \Phi(\cdot ; 1, t, xw, 1, 1/w)|{w=0} (dually \Phi(\cdot ; s, 1, 1, yw, 1/w)|{w=0}), and \Phi(\cdot ; 1/w, 1, x, y, w)|{w=0} (dually \Phi(\cdot ; 1, 1/w, x, y, w)|{w=0}) distinguish all SP posets with ten elements or fewer, but it remains inconclusive whether these specialization continue to distinguish SP posets of larger cardinality.

— A Generating Polynomial for Convex Subsets of Posets  (2610.23399 - George et al., 16 Oct 2026) in Section 5, “Posets Generated by + and ⊕”