research

2026

1. arXiv

   Graded Ehrhart theory for hypersimplices

   Nathaniel Libman, and Weston Miller

   2026

   Ehrhart theory counts lattice points in dilates of a polytope. Reiner and Rhoades conjectured a graded refinement, where that count is upgraded to a q-series coming from orbit harmonics. We prove rationality of this refinement for hypersimplices by writing down generators for the harmonics ideal, which as a bonus hands you the Hilbert series and the graded Frobenius characteristic. The combinatorial heart of it turns out to be a statement about 2-factors of regular multigraphs, and you can draw one to see the constructions run. The lattice-point counts underneath all of this are their own interactive page.

   Abs DOI Bib Website

   We prove that the q-Ehrhart series of a hyperplane slice of a cube is a rational function with an explicit denominator that satisfies q-reciprocity, confirming a conjecture of Reiner and Rhoades for these polytopes. To do this, we find a generating set for the orbit harmonics ideal, which also yields the Hilbert series and graded Frobenius characteristic of the associated quotient. We further show that the harmonic algebra of a hypersimplex ∆ is generated as an algebra by the harmonic space of ∆ using structural results on 2-factors of regular multigraphs. In particular, the harmonic algebra is finitely generated, giving a second proof of rationality.

   @article{gradedEhrhartHypersimplices,
     title = {Graded Ehrhart theory for hypersimplices},
     author = {Libman, Nathaniel and Miller, Weston},
     year = {2026},
     eprint = {2608.27438},
     archiveprefix = {arXiv},
     primaryclass = {math.CO},
     url = {https://arxiv.org/abs/2608.27438},
     doi = {10.48550/arXiv.2608.27438},
   }

2025

1. Alg. Comb.

   An elaborate new proof of Cayley’s formula

   Esther Banaian, Anh Trong Nam Hoang, Elizabeth Kelley, Weston Miller, Jason Stack, Carolyn Stephen, and Nathan Williams

   Algebraic Combinatorics, 2025

   Cayley's formula says there are nⁿ⁻² labelled trees on n vertices, and it has a dozen proofs that fit on a postcard. Ours goes through Deodhar components of a braid variety built from an affine Kac–Moody group of type A_n−1, Opdam's trace formula in the affine Hecke algebra, and an identity of Haglund. The bijection is completely explicit, though, so you can watch it run on a tree you pick yourself.

   Abs DOI Bib Website

   We construct a bijection between certain Deodhar components of a braid variety constructed from an affine Kac-Moody group of type A_n−1 and vertex-labeled trees on n vertices. By an argument of Galashin, Lam, and Williams using Opdam’s trace formula in the affine Hecke algebra and an identity due to Haglund, we obtain an elaborate new proof for the enumeration of the number of vertex-labeled trees on n vertices.

   @article{ALCO_2025__8_4_971_0,
     author = {Banaian, Esther and Hoang, Anh Trong Nam and Kelley, Elizabeth and Miller, Weston and Stack, Jason and Stephen, Carolyn and Williams, Nathan},
     title = {An elaborate new proof of {Cayley{\textquoteright}s} formula},
     journal = {Algebraic Combinatorics},
     pages = {971--995},
     publisher = {The Combinatorics Consortium},
     volume = {8},
     number = {4},
     year = {2025},
     doi = {10.5802/alco.429},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.429/},
   }

2. J. Algebra

   Rational Catalan numbers for complex reflection groups

   Weston Miller

   Journal of Algebra, 2025

   Galashin, Lam, Trinh, and Williams proved the enumeration of noncrossing Catalan objects for finite Coxeter groups uniformly, by evaluating the canonical symmetrizing trace on the Hecke algebra at a power of a Coxeter element. This paper runs the same machine on the spetsial complex reflection groups — the ones that behave as though they were Weyl groups of a group that does not exist — and rational Catalan numbers come back out the other end. There is a demo that computes this trace explicitly.

   Abs DOI Bib Code Poster Website

   Assuming standard conjectures, we show that the canonical symmetrizing trace on the Hecke algebra of irreducible spetsial complex reflection groups produces rational Catalan numbers when evaluated at powers of a Coxeter element. This extends a technique used by Galashin, Lam, Trinh, and Williams to uniformly prove the enumeration of their noncrossing Catalan objects for finite Coxeter groups.

   @article{MILLER202510,
     title = {Rational Catalan numbers for complex reflection groups},
     journal = {Journal of Algebra},
     volume = {672},
     pages = {10-30},
     year = {2025},
     issn = {0021-8693},
     url = {https://www.sciencedirect.com/science/article/pii/S0021869325000870},
     author = {Miller, Weston},
     keywords = {Reflection groups, Hecke algebras, Catalan numbers},
     doi = {10.1016/j.jalgebra.2025.01.027},
   }