Generating Function for the Final Descent

Abstract

The final descent statistic has a generating function determined by the Catalan series.

Theorem 1.1 (Final descent weights).

Lean statement: D5/S3/Combinatorics/InversionSeq/InversionSeq152FinalSeries.final_descent_enumeration

Proof. Machine-checked in Lean as D5/S3/Combinatorics/InversionSeq/InversionSeq152FinalSeries.final_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, q an element of R and C(x) the Catalan series with coefficients in R. Let H(x) have, as its coefficient of x to the power n, the sum of q to the power d over Dyck paths of semilength n, where d is the length of the final descent. The empty path has d equal to zero. Then (1 - q x C(x)) H(x) = 1.

References

• Truth anchor: D5/S3/Combinatorics/InversionSeq/InversionSeq152FinalSeries.final_descent_enumeration
• Dependency: D5/S3/Combinatorics/InversionSeq/InversionSeq152Factor