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Generating Function for the Penultimate Descent

Abstract

The weight of the penultimate descent has a generating function expressed through final descent weights.

Theorem 1.1 (Weighted penultimate descents).

Lean statement: D5/S3/Combinatorics/InversionSeq/InversionSeq152PenultimateSeries.penultimate_descent_enumeration

Proof. Machine-checked in Lean as D5/S3/Combinatorics/InversionSeq/InversionSeq152PenultimateSeries.penultimate_descent_enumeration (✓ 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 R be a commutative ring and w a function from nonnegative integers to R with w(0) = 0. Write C(x) for the Catalan series. In degree n, let P(x) sum w of the penultimate nonempty descent length over Dyck paths of semilength n, using length zero when there are fewer than two descents. Let F(x) sum w of the final descent length, using zero for the empty path. Then (1 - x) P(x) = x C(x) F(x).

References