  Buryak, Feigin and Nakajima computed a generating function for a family of
partition statistics by using the geometry of the $Z/cZ$ fixed point sets in
the Hilbert scheme of points on $C^2$. Loehr and Warrington had already shown
how a similar observation by Haiman using the geometry of the Hilbert scheme of
points on $C^2$ could be made purely combinatorial. We extend the techniques of
Loehr and Warrington to also account for cores and quotients. In particular, we
construct a multigraph $M_{r,s,c}$ that is a direct refinement of Loehr and
Warrington's multigraphs $M_{r,s}$, retains the relevant partition data, and is
preserved by an involution $I_{r,s,c}$ which we use to prove the
equidistribution of a family of partition statistics.
  As a consequence, we obtain a purely combinatorial proof of a result of
Buryak, Feigin, and Nakajima. 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 $c$,
  \begin{equation*}
  \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},
  \end{equation*}
  where the sum ranges over all partitions $\lambda$ with $c$-core $\mu$.
  Section 2 recalls background on partitions, cores and quotients and is
written with those new to the subject in mind.
