Ehrhart reciprocity and forced factors in three plane-partition enumerators

Abstract

We study three plane-partition enumerators arising from Schreier-Aigner's quasi-symmetry classes. Their realizations as lattice-point enumerators, together with staircase translations of interior lattice points, yield factorizations by Ehrhart-Macdonald reciprocity. We determine the consecutive linear factors, the parity and degree of the residual polynomials, and explicit divisibility bounds for their coefficient denominators. The denominator argument includes the half-integral translation required by the second-kind classes. We also derive a corrected size-five formula for the symmetric second-kind class. Exact computations verify irreducibility of the quasi-symmetric residual polynomials for every size from $3$ to $24$.