Simplices for Numeral Systems

Abstract

The family of lattice simplices in $\mathbb{R}^n$ formed by the convex hull of the standard basis vectors together with a weakly decreasing vector of negative integers include simplices that play a central role in problems in enumerative algebraic geometry and mirror symmetry. From this perspective, it is useful to have formulae for their discrete volumes via Ehrhart $h^\ast$-polynomials. Here we show, via an association with numeral systems, that such simplices yield $h^\ast$-polynomials with properties that are also desirable from a combinatorial perspective. First, we identify $n$-simplices in this family that associate via their normalized volume to the $n^{th}$ place value of a positional numeral system. We then observe that their $h^\ast$-polynomials admit combinatorial formula via descent-like statistics on the numeral strings encoding the nonnegative integers within the system. With these methods, we recover ubiquitous $h^\ast$-polynomials including the Eulerian polynomials and the binomial coefficients arising from the factoradic and binary numeral systems, respectively. We generalize the binary case to base-$r$ numeral systems for all $r\geq2$, and prove that the associated $h^\ast$-polynomials are real-rooted and unimodal for $r\geq2$ and $n\geq1$.