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Run Lengths under Dyck Path Reflection

Abstract

Reflection exchanges ascent lengths and descent lengths while reversing their order.

Theorem 1.1 (Reflected ascent lengths).

Lean statement: D5/S3/Combinatorics/InversionSeq/InversionSeq152Reflection.reflection_runs

Proof. Machine-checked in Lean as D5/S3/Combinatorics/InversionSeq/InversionSeq152Reflection.reflection_runs (✓ 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.

Reflect a Dyck path by reversing its step word and exchanging up steps with down steps. The list of lengths of its nonempty ascents is the reverse of the list of lengths of the original path’s nonempty descents.

References