Homotopical combinatorics, or, What equivariant ring spectra taught me about enumerating submonoids

Equivariant ring spectra come equipped with a class of multiplicative norm maps satisfying certain compatibility conditions. These form a factorization system on the subgroup lattice (considered as a category) and the assignment gives an equivalence between the homotopy category of G-N_\infty spectra and G-factorization systems. Homotopical combinatorialists have sought to use this correspondence to lift the combinatorics of factorization systems to equivariant ring spectra. In this talk, I will discuss how the homotopical viewpoint has led to a new result in combinatorics. Let M be a commutative monoid and let [n] be the monoid on {0,1,...n} with the max operation. We demonstrate how the enumeration of submonoids of M \times [n] (as a function of n) is amenable to the transfer matrix method. This results in a new derivation of Kaneko's poly-Bernoulli numbers. I will make my presentation accessible to both combinatorialists and topologists. This work is joint with the summer 2024 Reed Collaborative Mathematics Group.