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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2604.13238 |
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| _version_ | 1866918447112257536 |
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| author | Huang, Yifeng |
| author_facet | Huang, Yifeng |
| contents | We introduce a quadratic form $Q$ on the space of functions on the gap poset $G$ of the numerical semigroup $\langle a,b\rangle$. We prove combinatorially that when evaluated on the indicator function of an upward closed subset $D$, this quadratic form precisely recovers the Gorsky--Mazin $\mathtt{dinv}$ statistic of $D$, viewed as a Young subdiagram of $G$. Furthermore, we prove Theorem~1.2 that when evaluated on a pair of subdiagrams of $G$, the symmetric bilinear form associated with $Q$ is equal to a novel cross-$\mathtt{dinv}$ statistic, which is nonnegative. Combining these, we prove the inequality \[ Q(\mathbf{n})\geq \dfrac{1}{|G|}\,\|\mathbf{n}\|_\infty^2\] if $\mathbf{n}$ is a real-valued decreasing function on $G$, showing an effective positive definiteness of $Q$ on the corresponding cone. Theorem~1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13238 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A quadratic form generalization of rational dinv Huang, Yifeng Combinatorics We introduce a quadratic form $Q$ on the space of functions on the gap poset $G$ of the numerical semigroup $\langle a,b\rangle$. We prove combinatorially that when evaluated on the indicator function of an upward closed subset $D$, this quadratic form precisely recovers the Gorsky--Mazin $\mathtt{dinv}$ statistic of $D$, viewed as a Young subdiagram of $G$. Furthermore, we prove Theorem~1.2 that when evaluated on a pair of subdiagrams of $G$, the symmetric bilinear form associated with $Q$ is equal to a novel cross-$\mathtt{dinv}$ statistic, which is nonnegative. Combining these, we prove the inequality \[ Q(\mathbf{n})\geq \dfrac{1}{|G|}\,\|\mathbf{n}\|_\infty^2\] if $\mathbf{n}$ is a real-valued decreasing function on $G$, showing an effective positive definiteness of $Q$ on the corresponding cone. Theorem~1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver. |
| title | A quadratic form generalization of rational dinv |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.13238 |