# Verification report — OWR-16164-019 (Sportiello, Oberwolfach Report 23/2018)

Verification date: 2026-09-28.

**Verdict.** Complete negative answer. The conjecture "B_λ ≥ A_λ for every digitally convex shape λ" is false.
For the band λ = {(x,y) ∈ [7]² : −3 ≤ y−x ≤ 4}, A_λ = 536 > B_λ = 515. Unrefereed.

## Statement checked
- **Source.** Oberwolfach Reports 15 (2018), no. 2 (Report 23/2018, *Enumerative Combinatorics*), problem session,
  A. Sportiello, "A simple (?) problem in permutation patterns", pp. 1455–1457, DOI 10.4171/OWR/2018/23.
  - The source defines the patterns by pictures.
  - Pages 1456–1457 were rendered at high resolution and transcribed independently by the finder and by the
    referee. The transcriptions agree cell by cell.
  - The section text is identical in the MFO copy and in a fresh EMS Press copy.
- **Corpus record.** ulamai/UnsolvedMath, OWR-16164-019, status `open`. It is equivalent to the source, except
  that it attaches the red and blue crosses to the opposite colours. That swap is a bijection on colourings, so
  it does not change B_λ.

## The reading of the pictures
- **The adopted reading** (cells in λ, one common frame for shapes and patterns) is the only one tested that is
  consistent with all four of the report's own checks:
  - the worked 8×8 example (σ = 47231865 is in A_λ, and the printed colouring is in B_λ);
  - the drawn occurrence of a 4-point pattern;
  - the Remark (for the 4×4 shape, the identity is in A_λ but has no admissible colouring).
- **Alternatives:**
  - swapping the colours leaves the counts unchanged;
  - reading the patterns in a mirrored frame fails the checks and only mirrors the counterexample;
  - requiring the crosses to lie outside λ is the only reading under which no counterexample appears up to side 7,
    and it contradicts the text and all four checks.

## Computations (exact; scripts and outputs in reproducibility/)
- **The band:** |S_λ| = 1066, A_λ = 536, B_λ = 515, computed by three independent programs (finder, referee,
  lead auditor).
- **Census:**
  - All 349,320 digitally convex shapes of side ≤ 6 satisfy B ≥ A; the minimum positive ratio is 22/15.
  - Among the 5,693,968 shapes of side 7, exactly two violate the inequality: the band and its transpose.
  - The shape counts match Gessel's formula for convex polyominoes.
- **Bands of side 6–10:** the smallest ratio is 1.467, 0.961, 0.571, 0.336 and 0.184. The lead auditor
  recomputed sides 6, 8 and 9.
- **Context, proved in the paper:**
  - In the square, A = C_n and B = C(2n, n) = (n+1)A.
  - For shapes cut only at the top-left and bottom-right corners, B ≥ 2A.

## Independent adversarial audit
Verdicts (2026-09-28):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED_WITH_FIXES (the definitions are now given in coordinates and the load-bearing reading is identified) |
| Computation | CONFIRMED (reproduced three ways; the census was extended to side 7) |
| Answer as posed | CONFIRMED |
| Novelty | CONFIRMED, as far as can be checked |

## Relation to the literature, novelty and scope
- **Searches.** The arXiv API, OpenAlex, Crossref and web searches in September 2026 found no discussion of the
  conjecture. The searches covered the report's citing works and Sportiello's later papers.
- **Caveat.** The author may know privately that the conjecture fails. This negative search is not a proof of
  priority.
- **Not proposed.** The note does not propose a corrected conjecture.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/