The asymptotic number of zonotopes

Source abstract

We provide a sharp estimate for the asymptotic number of lattice zonotopes inscribed in [0, n] d as n tends to infinity. Our estimate refines the logarithmic equivalent established by B{á}r{á}ny, Bureaux, and Lund when the sum of the generators of the zonotope is prescribed. As we shall see, the exponential part of our estimate is a polynomial of degree d in n 1/(d+1) , plus a term involving the non-trivial zeros of the Riemann zeta function. Our analysis is based on a correspondence between power sums over primitive vectors (vectors with coprime coordinates) and Eulerian polynomials. We also analyse some combinatorial properties of lattice zonotopes. In particular, we compute the asymptotic mean and variance of the diameter of the graph of a random lattice zonotope inscribed in [0, n] d as n goes to infinity, and show that this diameter satisfies a central limit theorem.