bibkey: benyi2024pattern authors: Beáta Bényi, Toufik Mansour, José L. Ramírez year: 2024 title: “Pattern Avoidance in Weak Ascent Sequences” doi: 10.46298/dmtcs.12273 url: https://arxiv.org/abs/2309.06518v4 claim: “The sequence w_210(n) coincides with the sequence A117106.” strata_touched:

• D5/S3/Combinatorics/WeakAscent/WeakAscentSemiBaxter license: citation-only triage: anchor

Bényi, Mansour and Ramírez, pattern avoidance in weak ascent sequences

The paper enumerates weak ascent sequences avoiding a single pattern of length three for the patterns 001, 011, 012, 021 and 102, relates several of these classes to compositions, upper triangular 01-matrices and plane trees, and leaves the remaining length-three patterns open, with one conjectured enumeration.

Verified locator

DOI: 10.46298/dmtcs.12273

URL: https://arxiv.org/abs/2309.06518v4

• Locator: Section 1, a weak ascent sequence is a sequence e_1 ⋯ e_n of nonnegative integers with e_1 = 0 and e_i ≤ 1 + wasc(e_1 ⋯ e_{i−1}), where wasc counts the positions j with e_j ≤ e_{j+1}; w_p(n) counts those of length n avoiding the pattern p.
• Locator: Section 3, Table 2 lists w_p(n) for p ∈ {000, 010, 100, 101, 110, 120, 201, 210}, whose enumeration is left open.
• Locator: Section 3, Conjecture 3.1: the sequence w_210(n) coincides with A117106, which enumerates permutations avoiding the vincular pattern 2-41-3.

Reading of the statement

The numbers of 210-avoiding weak ascent sequences of length n = 0, …, 10 are 1, 1, 2, 6, 23, 104, 530, 2958, 17734, 112657, 750726; the same numbers count the permutations of [n] avoiding 2-41-3 (the semi-Baxter numbers of Bouvel, Guerrini, Rechnitzer and Rinaldi).

Bounded prior-resolution evidence

Read on 2026-10-01: the papers citing arXiv:2309.06518 that were located are Mansour, Three Classes of Pattern-Avoiding Weak Ascent Sequences (Experimental Mathematics, 2026; patterns 0123, 0012, 0021), Mansour, Statistics in Weak Ascent Sequences (Mathematics 14 (2026) 1378), Callan and Mansour on pairs and triples of patterns (Discrete Mathematics 348 (2025) 114438), and Zhou on revised ascent sequences (arXiv:2505.05171); none treats Conjecture 3.1. OEIS A117106 links the paper without recording a proof. This is a bounded negative finding.