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Relation Between Weighted Dyck Series

Abstract

The combined weight series is related to the downstep weight series.

Theorem 1.1 (Weighted series relation).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalGroundSeries.groundSeries_bridge

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalGroundSeries.groundSeries_bridge (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Sergi Elizalde, Amya Luo (2024). Pattern avoidance in nonnesting permutations. DOI: 10.48550/arXiv.2412.00336. URL: https://arxiv.org/abs/2412.00336v6.

Commentary.

Let G be the ordinary generating function of the combined weight sums, and let D be the ordinary generating function of the downstep weight sums. Then (1 - 2xD)G equals 1 - x.

References