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Recovering a Smaller Cycle from Insertion

Abstract

The prescribed boundary shape of an inverse image permits removal of three letters.

Theorem 1.1 (The recovered inner permutation).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateInsertionInverse.insertion_cycle_inverse

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateInsertionInverse.insertion_cycle_inverse (✓ 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 permutation w of size n at least five beginning with n whose inverse image begins with n, n minus one and ends with one, n minus two, there is a permutation q of size n minus three beginning with n minus three and ending with one such that I of the inverse image of q is the inverse image of w. That inverse image of q is a first-maximum permutation whose fundamental image and cycle from its maximum are q; if w avoids 132, then q avoids 132.

References