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FishburnTenSevenBStructure

Abstract

Permutations in the second triple-avoidance class have one of two interval forms.

Theorem 1.1 (Interval forms for the second class).

Lean statement: D5/S3/Combinatorics/FishburnTenSeven/FishburnTenSevenBStructure.B_interval_normalForm

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FishburnTenSeven/FishburnTenSevenBStructure.B_interval_normalForm (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Eric S. Egge (2022). Pattern-Avoiding Fishburn Permutations and Ascent Sequences. DOI: 10.48550/arXiv.2208.01484. URL: https://arxiv.org/abs/2208.01484v1.

Commentary.

For positive n, every Fishburn permutation of length n avoiding 1324, 2143 and 3124 has an m between one and n and one of two forms. The first is the decreasing interval from n through m plus one followed by a Fishburn permutation of length m avoiding 213 and starting with one. The second has two at most d, d at most h, and h less than m, and consists of the decreasing interval from n through m plus one, the decreasing interval from h through d, one, the increasing interval from h plus one through m minus one, m, and a Fishburn permutation of length d minus two avoiding 213 with each entry increased by one.

References