Enumeration of (32; 1 to 3) Avoiders

Abstract

The (32; 1 to 3) avoidance series satisfies a cubic equation and is its distinguished formal branch.

Theorem 1.1 (The cubic enumeration).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/ArrowThirtyTwoOneThree.result (✓ std3). ∎

Resolves. Problems/zhou-yu-arrow-32-13-enumeration (proved) by D5/S3/Combinatorics/ArrowThirtyTwoOneThree.result.

Source. Repository-derived.

Acknowledgement. Robin D.P. Zhou, Xinyang Yu (2026). Arrow-Wilf equivalences and enumerative results for short arrow patterns. DOI: 10.48550/arXiv.2609.29392. URL: https://arxiv.org/abs/2609.29392v1.

Commentary.

The generating series F of the (32; 1 to 3) avoidance numbers satisfies 1 + (3x - 2)F + (1 - x)(1 - 2x)F squared + x cubed F cubed = 0. It is the unique integer formal power series satisfying this equation with constant coefficient one and coefficient of x equal to one.

References

• Truth anchor: D5/S3/Combinatorics/ArrowThirtyTwoOneThree.result
• Dependency: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeAlgebra
• Dependency: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs
• Dependency: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeRefined
• Dependency: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeSeries