Trimer Covers in the Triangular Grid — Problem 12

Definitions and mathematical statement

Let $S_n$ denote the large Schröder numbers with the conventional normalization \[ S_0=1,\qquad S_n=S_{n-1}+\sum_{k=0}^{n-1}S_kS_{n-1-k}\quad(n\in{\mathbb{Z}},\ n\geq1). \] The supplied record uses the notation $T_{112}(a,b)$ from Propp's cited source but does not define it. In particular, the counted objects, the underlying region, the parameter conventions, and the meaning of the subscript $112$ are not supplied. These definitions must be recovered from the cited source before the question becomes self-contained.

Subject to that unresolved definition and confirmation that the source's phrase "$n$th large Schröder number" uses the normalization above, the proposed statement is \[ \forall n\in{\mathbb{Z}},\quad n\geq1\ \Longrightarrow\ T_{112}(n+2,2n+1)=S_n. \] The source record reports that equality has been verified for $1\leq n\leq15$; this is a supplied verification claim, not an independently checked result. The registry labels the problem open.

Short English statement

Propp asks whether $T_{112}(n+2,2n+1)$ equals the $n$th large Schröder number for every integer $n\geq1$. The supplied record reports agreement through $n=15$. A complete formulation requires the missing definition of $T_{112}$ and confirmation of the sequence indexing.

Sources

• Local UnsolvedMath registry, permanent ID $800353$, source code catalogue entry, title: Trimer Covers in the Triangular Grid --- Problem 12. The supplied record attributes the problem to Propp (2022) and labels it open; that status is supplied curation, not an independently verified literature assessment.
• Propp, Trimer Covers in the Triangular Grid: Twenty Mostly Open Problems (2022), source item Problem 12 (#5). Supplied source URL: https://arxiv.org/abs/2206.06472. The record reports access on 2026-10-16 and extraction by arxiv_latex; this provenance has not been independently verified here.
• OEIS entry A006318, referenced in the supplied statement for the large Schröder numbers: https://oeis.org/A006318.