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Indecomposability of Pattern Avoiders

Abstract

For 312- and 231-avoiders, indecomposability is characterized by an extreme endpoint.

Theorem 1.1 (Indecomposable 312-avoiders).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaBasicSumIndecomp.avoid312_indecomp_iff_last_one

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaBasicSumIndecomp.avoid312_indecomp_iff_last_one (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Kassie Archer, Robert P. Laudone (2024). Pattern avoidance and the fundamental bijection. DOI: 10.48550/arXiv.2407.06338. URL: https://arxiv.org/abs/2407.06338v1.

Commentary.

A nonempty 312-avoiding permutation is sum-indecomposable exactly when its last entry is one.

Theorem 1.2 (Indecomposable 231-avoiders).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaBasicSumIndecomp.avoid231_indecomp_iff_first_max

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaBasicSumIndecomp.avoid231_indecomp_iff_first_max (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Kassie Archer, Robert P. Laudone (2024). Pattern avoidance and the fundamental bijection. DOI: 10.48550/arXiv.2407.06338. URL: https://arxiv.org/abs/2407.06338v1.

Commentary.

A nonempty 231-avoiding permutation is sum-indecomposable exactly when its first entry is its maximum.

References