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Transitions of Active Marks

Abstract

Appending a letter induces record, fresh-site or old-site transitions on the active marks.

Theorem 1.1 (The active-mark transition correspondence).

Lean statement: D5/S3/Combinatorics/WeakAscent/WeakAscent215Transitions.left_transitions

Proof. Machine-checked in Lean as D5/S3/Combinatorics/WeakAscent/WeakAscent215Transitions.left_transitions (✓ 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.

For a nonempty word w in the class avoiding 100, 101, 110 and 201, put M equal to its maximum, b equal to one plus its weak ascent count minus M, and e true exactly when its last entry is M. Appendable letters are in bijection with either a gap g below b or a position s in the increasing active-value list. The gap choice appends M + g + 1, appends g false marks and a true mark, changes the budget to b minus g and sets true mode. A site choice appends its active value. At a false mark it truncates the mark list before s, retains budget b and sets false mode. At a true mark with e true and s last, it leaves the singleton true list, changes the budget to b + 1 and sets true mode. At any other true mark it leaves the empty list, retains budget b and sets false mode.

References