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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2208.09310 |
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| _version_ | 1866916109401194496 |
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| author | Vidalis, Eve |
| author_facet | Vidalis, Eve |
| contents | 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^{|λ|}t^{h_{x,c}^+(λ)}=q^{|μ|}\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 $λ$ with $c$-core $μ$.
Section 2 recalls background on partitions, cores and quotients and is written with those new to the subject in mind. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_09310 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A combinatorial proof of Buryak-Feigin-Nakajima Vidalis, Eve Combinatorics 05A17, 05E14 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^{|λ|}t^{h_{x,c}^+(λ)}=q^{|μ|}\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 $λ$ with $c$-core $μ$. Section 2 recalls background on partitions, cores and quotients and is written with those new to the subject in mind. |
| title | A combinatorial proof of Buryak-Feigin-Nakajima |
| topic | Combinatorics 05A17, 05E14 |
| url | https://arxiv.org/abs/2208.09310 |