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The Simplicial Geometry of Integer Partitions: An Exact $O(1)$ Formula via $A_{k-1}$ Root Systems

Antonio Bonelli

TL;DR

The paper tackles the exact evaluation of the partition function $p_k(n)$ by embedding the problem in the geometry of the $A_{k-1}$ Weyl chamber and applying a Simplicial Successive Decomposition (SSD). It establishes a unimodular triangulation into $N_k=\binom{k}{2}$ simplices, and derives an exact $O(1)$ closed-form formula using Brion's localization and a Betke-Kneser valuation framework, with residue shifts from a Residue Matrix $\mathcal{R}$. Key contributions include the Triangular Cardinality theorem, the SSD closed-form, Ehrhart–Macdonald reciprocity, and the Core Collapse regime, all supported by both theoretical proof and a $k=12$ experimental validation. The work provides a geometric, deterministic mechanism for partition enumeration, reducing the apparent combinatorial complexity to a fixed simplicial basis tied to the root system $A_{k-1}$, with practical implications for exact counting in high-dimension regimes where continuous approximations fail.

Abstract

We present a structural resolution to the exact evaluation of the partition function $p_k(n)$, addressing the limitations of traditional recursive and asymptotic methods. By introducing the Simplicial Successive Decomposition (SSD) framework, we demonstrate that the partition polytope $\mathcal{P}_{n,k}$ is not an arbitrary geometric object, but admits a rigid minimal unimodular triangulation into exactly $N_k = \binom{k}{2}$ simplices. This cardinality is determined by the positive root system of the $A_{k-1}$ Weyl chamber.We decompose Euler's generating function into a finite sum of simplicial rational transforms. By applying Brion's localization theorem and the negative binomial expansion, we derive an exact closed-form formula with $O(1)$ computational complexity. The validity of the model is confirmed through Ehrhart-Macdonald reciprocity, ensuring accuracy in the "Core Collapse" regime where the polytope's interior is empty and continuous volume approximations are inapplicable.

The Simplicial Geometry of Integer Partitions: An Exact $O(1)$ Formula via $A_{k-1}$ Root Systems

TL;DR

The paper tackles the exact evaluation of the partition function by embedding the problem in the geometry of the Weyl chamber and applying a Simplicial Successive Decomposition (SSD). It establishes a unimodular triangulation into simplices, and derives an exact closed-form formula using Brion's localization and a Betke-Kneser valuation framework, with residue shifts from a Residue Matrix . Key contributions include the Triangular Cardinality theorem, the SSD closed-form, Ehrhart–Macdonald reciprocity, and the Core Collapse regime, all supported by both theoretical proof and a experimental validation. The work provides a geometric, deterministic mechanism for partition enumeration, reducing the apparent combinatorial complexity to a fixed simplicial basis tied to the root system , with practical implications for exact counting in high-dimension regimes where continuous approximations fail.

Abstract

We present a structural resolution to the exact evaluation of the partition function , addressing the limitations of traditional recursive and asymptotic methods. By introducing the Simplicial Successive Decomposition (SSD) framework, we demonstrate that the partition polytope is not an arbitrary geometric object, but admits a rigid minimal unimodular triangulation into exactly simplices. This cardinality is determined by the positive root system of the Weyl chamber.We decompose Euler's generating function into a finite sum of simplicial rational transforms. By applying Brion's localization theorem and the negative binomial expansion, we derive an exact closed-form formula with computational complexity. The validity of the model is confirmed through Ehrhart-Macdonald reciprocity, ensuring accuracy in the "Core Collapse" regime where the polytope's interior is empty and continuous volume approximations are inapplicable.
Paper Structure (14 sections, 9 theorems, 3 equations, 3 figures, 4 tables)

This paper contains 14 sections, 9 theorems, 3 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Foundational Geometric Properties: Total Unimodularity
  3. Polyhedral Theory and the Triangular Basis
  4. The Betke-Kneser Valuation Ring
  5. Theorem: The Triangular Cardinality of Simplicial Basis
  6. Derivation of the SSD Closed-Form Formula
  7. Stability and Reciprocity
  8. The Core Collapse Theorem
  9. Experimental Results: $k=12$ Stress Test
  10. Comparative Regime Analysis
  11. Structural Implications: The Geometry of the Simplex
  12. Conclusion
  13. Python Implementation for $k=6$
  14. Technical Appendix: The Residue Matrix $\mathcal{R}_6$

Key Result

Theorem 2.1

The constraint matrix $A$ defining the Weyl chamber $\mathcal{C}_k = \{x \in \mathbb{R}^k : 1 \le x_1 \le \dots \le x_k\}$ is totally unimodular.

Figures (3)

  • Figure 1: Unimodular triangulation of the $A_{k-1}$ Weyl chamber into fundamental simplices.
  • Figure 2: Visual representation of the Simplicial Layers. The value $m_i$ dictates the depth of the lattice point layer within the simplex $\sigma_i$.
  • Figure 3: Geometric comparison between the Stable Regime (interior points exist) and the Core Collapse (interior is empty, points restricted to boundary).

Theorems & Definitions (17)

  • Theorem 2.1: Total Unimodularity
  • proof
  • Lemma 2.2: Modular Isomorphism
  • proof
  • Lemma 3.1: Basis Expansion Lemma
  • proof
  • Theorem 3.2: Triangular Decomposition Law
  • proof
  • Theorem 3.3
  • proof
  • ...and 7 more