A combinatorial proof of Buryak-Feigin-Nakajima

TL;DR

This paper provides a purely combinatorial proof of a generating function formula for partition statistics related to cores and quotients, extending previous geometric and combinatorial methods.

Contribution

It introduces a refined multigraph and involution to prove equidistribution of partition statistics, offering a new combinatorial proof of a result by Buryak, Feigin, and Nakajima.

Findings

Constructed a multigraph $M_{r,s,c}$ that refines Loehr and Warrington's graphs.

Proved the equidistribution of a family of partition statistics using an involution.

Derived a generating function formula for partitions with fixed cores.

Abstract

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.