A Counterexample to a Conjecture of Sportiello on Coloured Permutations in Convex Shapes

At the 2018 Oberwolfach workshop on enumerative combinatorics, A. Sportiello compared two families of objects attached to a digitally convex shape λ (a convex polyomino): the permutations whose graph lies in λ and that avoid a 123-pattern whose two corner cells also lie in λ, counted by A_λ, and the red–blue coloured permutations in λ that avoid two coloured versions of the 12-pattern, counted by B_λ. For the full square these numbers are the Catalan number and the central binomial coefficient, and he conjectured that B_λ ≥ A_λ for every digitally convex shape. We show that the conjecture is false. For the band λ = {(x,y) ∈ [7]² : −3 ≤ y−x ≤ 4} one has A_λ = 536 and B_λ = 515. An exhaustive computation shows that every digitally convex shape of side at most 6 satisfies the inequality, and that among the 5,693,968 shapes of side 7 exactly this band and its transpose violate it. For bands of side 10 the ratio B_λ/A_λ drops to about 0.18. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-16164-019. Paper page: https://eulersolve.org/papers/owr-16164-019/