publications

2025

1. Rational Catalan numbers for complex reflection groups

   Weston Miller

   Journal of Algebra, 2025

   Abs DOI Bib Code Poster

   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}
   }

2. 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

   Abs DOI 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{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/}
   }