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Buryak, Feigin and Nakajima computed a generating function for a family ofpartition 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 andWarrington had already shown how a similar observation by Haiman using thegeometry of the Hilbert scheme of points on $\mathbb{C}^2$ could be made purelycombinatorial. We extend Loehr and Warrington's techniques to also account forcores and quotients. As a consequence, we obtain a purely combinatorial proofof a generalisation 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 positiveintegers $c$,  \begin{equation*}  \sum q^{|\lambda|}t^{h_{x,c}^+(\lambda)}=q^{|\mu|}\prod_{i\geq1}\frac{1}{(1-q^{ic})^{c-1}}\prod_{j\geq 1}\frac{1}{1-q^{jc}t},  \end{equation*}  where the sum ranges over all partitions $\lambda$ with $c$-core $\mu$.

A combinatorial proof of Buryak-Feigin-Nakajima