publications

2024

1. 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

   arXiv preprint arXiv:2402.07798, 2024

   Abs arXiv Bib

   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{banaian2024elaborate,
     title = {An elaborate new proof of Cayley's formula},
     author = {Banaian, Esther and Hoang, Anh Trong Nam and Kelley, Elizabeth and Miller, Weston and Stack, Jason and Stephen, Carolyn and Williams, Nathan},
     journal = {arXiv preprint arXiv:2402.07798},
     year = {2024}
   }

2023

1. Rational Catalan Numbers for Complex Reflection Groups

   Weston Miller

   arXiv preprint arXiv:2310.12354, 2023

   Abs arXiv Bib Code Poster

   Assuming standard conjectures, we show that the canonical symmetrizing trace evaluated at powers of a Coxeter element produces rational Catalan numbers for irreducible spetsial complex reflection groups. 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{miller2023rational,
     title = {Rational Catalan Numbers for Complex Reflection Groups},
     author = {Miller, Weston},
     journal = {arXiv preprint arXiv:2310.12354},
     year = {2023}
   }