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Reduction along the Distinguished Cycle

Abstract

The first two fundamental images of the distinguished cycle reduce avoidance to conditions on the parameter.

Theorem 1.1 (Images and avoidance of the cycle family).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateCycleReduction.P_cycle_reduction

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateCycleReduction.P_cycle_reduction (✓ 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.

For a nonempty parameter permutation r of length h, the first image of P(r) is h plus two followed by r increased by one and then one, and the second image is the fundamental image of r increased by one followed by h plus two and one. Avoidance through the second iterate is equivalent to 132-avoidance of r, its fundamental image and b(r), together with the absence of an increasing positional pair in b(r) straddling the first value of r plus one. The first and last values of P(r) are h plus two and the first value of r plus one, and its values at positions labelled by r are the corresponding shifted successors.

References