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The first and third values before a late minimum

Abstract

In a simple permutation in C whose minimum has zero-based position at least three, the first entry is exactly one more than the third entry.

Theorem 1.1 (The first and third values before a late minimum).

Lean statement: D5/S3/Combinatorics/PopStack/PopStackThirdEntry.third_entry_before_minimum

Proof. Machine-checked in Lean as D5/S3/Combinatorics/PopStack/PopStackThirdEntry.third_entry_before_minimum (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Lapo Cioni, Luca Ferrari, Rebecca Smith (2025). Sorting permutations using a pop stack with a bypass. DOI: 10.1016/j.disc.2025.114964. URL: https://arxiv.org/abs/2503.08285v1.

Commentary.

In a simple permutation in C whose minimum has zero-based position at least three, the first entry is exactly one more than the third entry.

References