The Bivariate Pure-History Series

Abstract

The pure-history series records both expenditure and length.

Definition 1.1 (The pure-history series).

Lean statement: D5/S3/Combinatorics/WeakAscent/WeakAscent215Products.pureSeries

Formalization. D5/S3/Combinatorics/WeakAscent/WeakAscent215Products.pureSeries (✓ std3).

Source. Repository-derived.

Acknowledgement. David Callan, Toufik Mansour (2025). Ascent Sequences and Weak Ascent Sequences Avoiding a Quadruple of Length-3 Patterns. DOI: 10.5281/zenodo.17144266. URL: https://math.colgate.edu/~integers/z80/z80.pdf.

Commentary.

The coefficient of z to the power s and x to the power l in P(z,x) is the number of pure histories of expenditure s and length l. Coefficients are nonnegative integers.

References

• Truth anchor: D5/S3/Combinatorics/WeakAscent/WeakAscent215Products.pureSeries
• Dependency: D5/S3/Combinatorics/WeakAscent/WeakAscent215Finite
• Dependency: D5/S3/Combinatorics/WeakAscent/WeakAscent215Pieces