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The (32; 1 to 3) Avoidance Series

Abstract

The avoidance numbers and their ordinary generating series are defined for the arrow pattern (32; 1 to 3).

Definition 1.1 (Avoidance numbers).

Lean statement: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs.count

Formalization. D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs.count (✓ std3).

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 number at n is the cardinality of the permutations of 1 through n avoiding (32; 1 to 3).

Definition 1.2 (Ordinary generating series).

Lean statement: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs.series

Formalization. D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs.series (✓ std3).

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 coefficient of x to the n in this integer power series is the avoidance number at n.

Definition 1.3 (Cubic equation and distinguished branch).

Lean statement: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs.claim

Formalization. D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs.claim (✓ std3).

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 avoidance series F satisfies 1 + (3x - 2)F + (1 - x)(1 - 2x)F squared + x cubed F cubed = 0. Any integer formal power series G satisfying the same equation, with constant coefficient one and coefficient of x equal to one, equals F.

References

  • Truth anchor: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs.claim
  • Truth anchor: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs.count
  • Truth anchor: D5/S3/Combinatorics/ArrowThirtyTwoOneThreeDefs.series
  • Dependency: D5/S3/Combinatorics/ArrowWilfDefs