demos
Interactive companions to my papers, and a playground for the theory behind them. Each one runs a construction on an example you choose.
Hypersimplex harmonics
Companion to Graded Ehrhart theory for hypersimplices (arXiv, 2026). Draw a loopless multigraph. The app writes it as a polynomial of matchings in the harmonic algebra or as a linear combination of maximal tableaux. The app can also compute the rational form of the q-Ehrhart series for any slice of the cube.
Cayley's formula
Companion to An elaborate new proof of Cayley's formula (Algebraic Combinatorics, 2025). Pick a vertex-labelled tree on n vertices and see it three ways at once: as a cyclic embedding, as the cyclic factorization read off by walking clockwise from vertex n, and as the corresponding maximal distinguished subword.
Rational Catalan numbers
Companion to Rational Catalan numbers for complex reflection groups (J. Algebra, 2025). Choose a complex reflection group and parameter. The app computes the canonical symmetrizing trace on the Hecke algebra at a power of a Coxeter element and computes the corresponding rational Catalan number. There are also examples in type A showing how this counts rational Dyck paths.
Ehrhart theory
Dilate a polytope, count the lattice points, and a polynomial appears. Pick a slice of the cube in two or three dimensions, or draw a lattice polygon of your own, and the app gives you the counting function, the h*-vector, the rational form of the Ehrhart series, and a check of Ehrhart–Macdonald reciprocity. Where it can, it also computes the graded q-refinement from Graded Ehrhart theory for hypersimplices.