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Occurrences of Length-Three Patterns

Abstract

Occurrences of length-three patterns are characterized by three positions and their pairwise comparisons.

Theorem 1.1 (Three-position characterization).

Lean statement: D5/S3/Combinatorics/InversionSeq/InversionSeqOccurs.occurs_three_iff

Proof. Machine-checked in Lean as D5/S3/Combinatorics/InversionSeq/InversionSeqOccurs.occurs_three_iff (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. David Callan, Toufik Mansour (2023). Inversion Sequences Avoiding Quadruple Length-3 Patterns. DOI: 10.5281/zenodo.8399694. URL: https://math.colgate.edu/~integers/x78/x78.pdf.

Commentary.

Let a, b and c be positive integers containing every rank from one through their maximum. The pattern with entries a, b and c occurs in a word if and only if there are three strictly increasing positions in the word whose entries have exactly the same pairwise strict inequalities and equalities as a, b and c.

References