The Fishburn Condition under Direct Sum

Abstract

The Fishburn condition is preserved and reflected by direct sums with separated values.

Theorem 1.1 (Componentwise Fishburn condition).

Lean statement: D5/S3/Combinatorics/FishburnTenThirteen/FishburnBasicSum.isFishburn_directSum_iff

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FishburnTenThirteen/FishburnBasicSum.isFishburn_directSum_iff (✓ 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.

Let m be a nonnegative integer, let every entry of a word u be at most m, and let every entry of a word v be positive. The concatenation of u with the word obtained by increasing every entry of v by m is Fishburn if and only if both u and v are Fishburn.

References

• Truth anchor: D5/S3/Combinatorics/FishburnTenThirteen/FishburnBasicSum.isFishburn_directSum_iff
• Dependency: D5/S3/Combinatorics/Fishburn/FishburnBasicInsertion
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicSum