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Active Values and Occurrence Marks

Abstract

Increasing active values and their occurrence marks associate a Boolean stack with a word.

Definition 1.1 (The increasing active-value list).

Lean statement: D5/S3/Combinatorics/WeakAscent/WeakAscent215WordHistory.activeValues

Formalization. D5/S3/Combinatorics/WeakAscent/WeakAscent215WordHistory.activeValues (✓ 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 word w, the active-value list consists, in increasing order, of the nonnegative values at most its maximum that can be appended to give a weak ascent sequence of length one more avoiding 100, 101, 110 and 201. The maximum of the empty word is taken to be zero.

Definition 1.2 (Occurrence marks of active values).

Lean statement: D5/S3/Combinatorics/WeakAscent/WeakAscent215WordHistory.activeMarks

Formalization. D5/S3/Combinatorics/WeakAscent/WeakAscent215WordHistory.activeMarks (✓ 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 active-mark list replaces each entry of the increasing active-value list by true when that value occurs in the original word and by false otherwise.

References