Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2603.21221

The Partition Graph as a Growing Discrete Geometric Object

Fedor B. Lyudogovskiy
View original source ↗Read source document ↗Export metadata ↓

Abstract

For each positive integer $n$, let $G_n$ be the graph of integer partitions of $n$, where two partitions are adjacent if one is obtained from the other by an elementary transfer of a cell in the Ferrers diagram, followed by reordering. Previous work has studied the global homotopy type of the clique complex $Cl(G_n)$ and the local combinatorics of $G_n$ at a fixed vertex. This paper initiates the study of $G_n$ itself as a growing discrete geometric object. It introduces a structural language for the large-scale morphology of partition graphs, centered on the antenna vertices, main chain, boundary framework, self-conjugate axis, simplex layers, degree landscape, central region, and spine. Using local invariants from the companion local theory, it also defines canonical vertex layerings of $G_n$. A small computational atlas for $1 \le n \le 12$ is included to illustrate how these structures emerge and interact. The paper is intended as a foundational and exploratory contribution, providing a vocabulary, a first structural picture, and a set of open directions for future quantitative and asymptotic work.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fedor B. Lyudogovskiy. 2026-03-22. The Partition Graph as a Growing Discrete Geometric Object. https://arxiv.org/abs/2603.21221

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. Liu proposed the conjecture \[ \sum_{\text{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \ge 2+\left(\frac{2r}{R}\right)^k,\qquad k>1, \] with the reverse inequality for $k<1$. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.

math.GM↗
arXiv2026

A quadratic critical-value conjecture for the fifth Bessel moment

We conjecture an explicit evaluation of the pure fifth Bessel moment $\int_0^\infty K_0(t)^5\,dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\mathbb{Q}(\sqrt{5})$ and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove three exact modular identities for $f$: the Petersson-norm formula $\langle f,f\rangle_{60} = \frac{3(5-\sqrt{5})}{2π^4}|L(f,2)|^2$, the coefficient-conjugation relation $L(f^σ,2) = κL(f,2)$ with explicit $κ\in \mathbb{Q}(\sqrt{5},i)$, and the twisted symmetric-square evaluation $L(χ_{-4}\mathrm{Sym}^2 f,2) = \sqrt{15}\,π^2 \langle f,f\rangle_{60}$, together with $L(χ_{-4}\mathrm{Sym}^2 f,3) = π^4\langle f,f\rangle_{60}/8$, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture $D_{5,\mathrm{odd}} = \frac{3\sqrt{15}(5-\sqrt{5})}{2}|L(f,2)|^2$ is equivalent to the symmetric-square conjecture $D_{5,\mathrm{odd}} = π^2 L(χ_{-4}\mathrm{Sym}^2 f,2)$ of Lim, Tu and Yu, while the exact relation $D_{5,\mathrm{even}} = π^2 D_{5,\mathrm{odd}}/(2\sqrt{15})$ follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.

math.GM↗