Symmetries of generalized cluster complexes via an extended cyclic sieving phenomenon. (with Cati Alexandru)
We study symmetries of the type A generalized cluster complex ∆⁽ᵐ⁾(n), a simplicial complex whose faces correspond to noncrossing dissections of a convex (mn+2)-gon by m-divisible diagonals, and which is topologically a wedge of spheres. We analyze the action of polygon symmetries f —rotations and reflections— on its top homology and compute the associated Lefschetz numbers L(m,n;f) using Hopf trace formulas. We give explicit, closed expressions for the L(m,n;f), and realize them as polynomial evaluations of the q,t-Catalan numbers at certain roots of unity. This construct extends the notion of the cyclic sieving phenomenon (CSP), from permutation actions to arbitrary ones.
Our results are obtained through parity analysis, binomial identities, and combinatorial arguments. Further, we provide evidence for broader symmetry patterns in these complexes, with connections to q-Kirkman polynomials and the combinatorics of the shuffle theorem.
Hurwitz numbers and transportation polytopes: regular cycle types. (with Isaac Berger)
We study the Hurwitz numbers H_g(), which enumerate transitive factorizations in S_n of a permutation of cycle type λ ⊢n into transpositions, equivalently counting branched covers of the Riemann sphere by genus-g surfaces. Although the ELSV formula connects H_g(λ) to intersection theory on the moduli space of curves, explicit closed forms are known only in special cases. We focus on the regular types λ=(n,n) and (n,n,1) considering the series
H(λ;t)=∑_{₉≥₀} H_g(λ)*t⁽ⁿ⁺ᶜʸᶜ⁽λ⁾⁻²⁺²ᵍ⁾/(n+cyc(λ)-2+2g)! .
Using symmetric function identities and a parametrization of the relevant characters –those that are non-zero on λ–, we derive explicit formulas for H((n,n);t) and H((n,n,1);t), expressed as a product of elementary exponential factors and a finite sum involving central binomial coefficients. Beyond the closed form itself, we relate the resulting expression to the h-vector of a central transportation polytope, and conjecture a general such correspondence for all regular cycle types.
This project started as a summer 2024 REU at Brandeis University.
Isaac gave a talk on the project at JMM Seattle.
An article is in preparation.
Cyclic sieving for block factorizations of the long cycle. (with Justin Bailey)
Hurwitz knew already in 1889 that there are n^{n-2} minimum length factorizations of the long cycle (123...n) of the symmetric group in transpositions; the same number that counts labeled trees on n vertices. If instead of transpositions, we consider cycles of fixed length, then there are n^{c-1} such factorizations with c factors; this is a special case of the Goulden-Jackson cactus formula.
We study a cyclic action on the set of such factorizations and we count its orbits of different sizes. Our answer is in the form of a cyclic sieving phenomenon (CSP), which means that these orbit sizes are given as polynomial evaluations at roots of unity.
This project started as a summer 2021 REU at UMass Amherst.
Justin gave a talk about the project at the UCONN REU Virtual Conference.
Justin's REU report and poster for the Goulden-Jackson retirement conference Combinatorial and Algebraic Enumeration are available.