Facet normals of Minkowski sums of staircase simplices

Abstract

Staircase simplices are the simplices of the standard (Freudenthal--Kuhn) triangulation of $[0,1]^d$ and of Stanley's triangulations of order polytopes. We study the facet normals of their Minkowski sums and how the number of summands constrains the size of primitive integer normals. We characterize the normals attainable with a prescribed number of summands in terms of $d-1$ linearly independent zero-one vectors orthogonal to the normal, whose supports have an intersection graph that can be colored with one color per summand. This gives a combinatorial characterization of the least number of summands needed to realize a normal. For two summands, all primitive normals have entries in $\{-1,0,1\}$, and we determine which of them occur. For three summands we bound every entry in absolute value by $3^{(d-2)/3}$ and construct normals with entries of order $θ^d$, $θ$ the plastic number; for four summands a second construction gives entries of order $ψ^d$, $ψ$ the supergolden ratio. Since $ψ>3^{1/3}$, the largest coefficients for three and four summands are exponentially separated. For vectors with at most two distinct nonzero entry values, we determine the least number of summands exactly, using a matching-rank formula proved by exterior algebra and representations of symmetric groups. With no restriction on the number of summands, the maximum coefficient equals the classical maximum for facets of $0/1$-polytopes in the same dimension.