GENERATING FUNCTIONS AND TRIANGULATIONS FOR LECTURE HALL CONES

Abstract

We investigate the arithmetic-geometric structure of the lecture hall cone L-n :- {lambda is an element of R-n : 0 <= lambda(1)/1 <= lambda(2)/2 <= lambda(3)/3 <= ... <= lambda(n/)n} We show that L-n is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart h*-polynomial is given by the (n-1) st Eulerian polynomial and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for L-n, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of L-n, including connections between enumerative and algebraic properties of L-n and cones over unit cubes.