A Combinatorial Proof of Buryak-Feigin-Nakajima

Abstract

Buryak, Feigin and Nakajima computed a generating function for a family of partition statistics by using the geometry of the $\mathbb{Z}/c\mathbb{Z}$ fixed point sets in the Hilbert scheme of points on $\mathbb{C}^2$. Loehr and Warrington had already shown how a similar observation by Haiman using the geometry of the Hilbert scheme of points on $\mathbb{C}^2$ can be made purely combinatorial. We extend Loehr and Warrington's techniques to also account for cores and quotients. As a consequence, we obtain a purely combinatorial proof of Buryak, Feigin, and Nakajima's result.

More precisely, we define a family of partition statistics $\{h_{x,c}^+, x\in (0,\infty]\}$ and give a combinatorial proof that for all $x$ and all positive integers $c$,
$$
\sum q^{|\lambda|}t^{h_{x,c}^+(\lambda)}=q^{|\mu|}\prod_{i\geq 1}\frac{1}{(1-q^{ic})^{c-1}}\prod_{j\geq 1}\frac{1}{1-q^{jc}t},
$$
where the sum ranges over all partitions $\lambda$ with $c$-core $\mu$.