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Bounds from the Parameter Cycle

Abstract

Avoidance in a parameter, its successor word, and its cycle word controls the terminal value and low suffix.

Theorem 1.1 (A lower bound for the terminal value).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateCycleScan.terminal_value_at_least_half

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateCycleScan.terminal_value_at_least_half (✓ 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.

Let r and q be permutations of the same size h at least four, with q beginning at h and its cyclic successor map equal to the permutation map of r. If r starts with h, h minus one, has penultimate value one and terminal value v at least two, and r, b(r), and q all avoid 132, then h is at most twice v.

Theorem 1.2 (The decreasing low suffix).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateCycleScan.low_suffix_descending

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateCycleScan.low_suffix_descending (✓ 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.

Under the same cycle and avoidance conditions, if v is below h minus two, then the entry of r at each position i from v through h minus three is h minus one minus i.

References