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Prepending a new second-smallest entry

Abstract

Let p have distinct positive entries and minimum one in its second position. Prepend two and increase every original entry other than one by one. The resulting word belongs to C if and only if p belongs to C.

Theorem 1.1 (Prepending a new second-smallest entry).

Lean statement: D5/S3/Combinatorics/PopStack/PopStackLeft.prepend_two_inC

Proof. Machine-checked in Lean as D5/S3/Combinatorics/PopStack/PopStackLeft.prepend_two_inC (✓ 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.

Let p have distinct positive entries and minimum one in its second position. Prepend two and increase every original entry other than one by one. The resulting word belongs to C if and only if p belongs to C.

References