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The Second Avoidance Class

Abstract

The positive entries and intervening zeros characterize the second Class 215 family.

Theorem 1.1 (An intrinsic avoidance criterion).

Lean statement: D5/S3/Combinatorics/WeakAscent/WeakAscent215Right.intrinsic_right_iff

Proof. Machine-checked in Lean as D5/S3/Combinatorics/WeakAscent/WeakAscent215Right.intrinsic_right_iff (✓ 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 of nonnegative integers whose entry at position zero is zero, using zero also for a missing entry, avoidance of 021, 101, 201 and 210 is equivalent to both of the following conditions. The positive entries in their original order are weakly increasing. Whenever a positive entry precedes a zero and that zero precedes a later entry, the first and last of these three entries are unequal. The empty word is included.

Theorem 1.2 (Appending in the second class).

Lean statement: D5/S3/Combinatorics/WeakAscent/WeakAscent215Right.right_append_iff

Proof. Machine-checked in Lean as D5/S3/Combinatorics/WeakAscent/WeakAscent215Right.right_append_iff (✓ 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.

Let w be a nonempty weak ascent sequence avoiding 021, 101, 201 and 210. A nonnegative letter a can be appended while preserving this class exactly when a is at most one plus the weak ascent count of w and either a is zero, a is greater than the maximum of w, or a equals that maximum and the last entry of w is positive.

References