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A Consecutive Pair Forced by a Return

Abstract

A terminal one after two inverse applications determines an adjacent pair in the original permutation.

Theorem 1.1 (The pair before one).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaCube231Pair.terminal_inverse_edge_forces_pair

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaCube231Pair.terminal_inverse_edge_forces_pair (✓ 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 of size n greater than one starting with n and ending with c, if its second inverse image ends with one, then c is below n and c plus one occurs immediately before one.

References