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Generating Function for a Marked Interior Descent

Abstract

The sum of weights on interior descents is related to the weight on the final descent.

Theorem 1.1 (Weighted interior descents).

Lean statement: D5/S3/Combinatorics/InversionSeq/InversionSeq152MarkedSeries.marked_interior_descent_enumeration

Proof. Machine-checked in Lean as D5/S3/Combinatorics/InversionSeq/InversionSeq152MarkedSeries.marked_interior_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 M(x) sum, over Dyck paths of semilength n, the weights w of the lengths of all nonempty descents except the last, and let F(x) sum the weight w of the final descent length. The empty path has final descent length zero. Then (2 - C(x)) M(x) = (C(x) - 1) F(x).

References