Schröder Enumeration of the Sorting Class of the Stack Avoiding 23-1

Zhao, arXiv:2410.17057v1, Conjecture 3.30 (Section 3.2, printed page 11):

The sorting class of SC_{23̲1} is enumerated by |Sort_n(SC_{23̲1})| = S_{n−1}.

SC_{23̲1} is the right-greedy stack map whose stack, read from top to bottom, must avoid the vincular pattern 23̲-1 (the entries playing 2 and 3 adjacent); Sort_n(SC_{23̲1}) is the set of permutations of length n whose image avoids 231, and S_m is the large Schröder number.

The theorem D5/S3/Combinatorics/VincularStack/VincularStackSort.result establishes the statement: for every n ≥ 1 the sorting class has Nat.largeSchroder (n − 1) elements.

Pre-registration issue 12117 records the literature screen: the only work citing arXiv:2410.17057 concerns the fully consecutive pattern, and the arXiv, OEIS and GitHub searches located no treatment of this conjecture. This is a bounded negative finding.

1. The sortable permutations are characterized directly from the stack rule by avoidance of an explicit mesh pattern.
2. Inserting a new maximum gives a generating tree with root label 2 and succession rule (k) → (3)(4)⋯(k)(k + 1)(k + 1), with explicit inverse operations.
3. The rule gives the functional equation R = 1 + xR + xR^2 of the large Schröder numbers.

The statement would fail if the mesh characterization missed a sortable permutation or admitted an unsortable one, or if an insertion produced a child with the wrong label or failed to be reversible.

The characterization, the generating tree and its inverse were checked on all permutations of length at most 11; the counts agree with S_{n−1} through n = 11.

theorem; the statement is Conjecture 3.30 of arXiv:2410.17057 and is quantified over every n ≥ 1.

The literature screen is limited to the citation index, arXiv, OEIS and GitHub searches and the repository checks recorded above; the journal version was not read in full.