Three-Vertex Weighted Bond Mobius Computation

Abstract

The literal three-vertex weighted bond posets have explicit Mobius polynomials.

Theorem 1.1 (The triangle source polynomial).

$$sourceMobiusPolynomial\left(triangleThree\right)=2+5\cdot X+2\cdot X^2$$

Proof. Machine-checked in Lean as D5/S0/Certificates/GonzalezDLeonWachsThreeVertexMobius.triangle_three_source_mobius_polynomial (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Rafael S. Gonzalez D’Leon; Michelle L. Wachs (2026). Weighted bond posets and a new chromatic symmetric function. DOI: 10.48550/arXiv.2608.08692. URL: https://arxiv.org/html/2608.08692v1#S4.Thmtheorem13.

Commentary.

The proof classifies every connected weighted partition of K3, identifies the actual order intervals, and evaluates the incidence-algebra recurrence.

Theorem 1.2 (The path source polynomial).

$$sourceMobiusPolynomial\left(pathThree\right)=1+3\cdot X+1\cdot X^2$$

Proof. Machine-checked in Lean as D5/S0/Certificates/GonzalezDLeonWachsThreeVertexMobius.path_three_source_mobius_polynomial (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Rafael S. Gonzalez D’Leon; Michelle L. Wachs (2026). Weighted bond posets and a new chromatic symmetric function. DOI: 10.48550/arXiv.2608.08692. URL: https://arxiv.org/html/2608.08692v1#S4.Thmtheorem13.

Commentary.

The same source-faithful classification for P3 retains exactly its two graph edges and evaluates the actual maximal weighted-partition sum.

References

• Truth anchor: D5/S0/Certificates/GonzalezDLeonWachsThreeVertexMobius.path_three_source_mobius_polynomial
• Truth anchor: D5/S0/Certificates/GonzalezDLeonWachsThreeVertexMobius.triangle_three_source_mobius_polynomial
• Dependency: D5/S0/Certificates/GonzalezDLeonWachsWeightedBondSource