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 \(\lambda\) (a convex polyomino): the permutations whose graph lies in \(\lambda\) and that avoid a \(123\)-pattern whose two corner cells also lie in \(\lambda\), counted by \(A_\lambda\), and the red–blue coloured permutations in \(\lambda\) that avoid two coloured versions of the \(12\)-pattern, counted by \(B_\lambda\). For the full square these numbers are the Catalan number and the central binomial coefficient, and he conjectured that \(B_\lambda\ge A_\lambda\) for every digitally convex shape. We show that the conjecture is false. For the band \(\lambda=\{(x,y)\in[7]^2:-3\le y-x\le 4\}\) one has \(A_\lambda=536\) and \(B_\lambda=515\). 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 \(B_\lambda/A_\lambda\) drops to about \(0.18\). This is an unrefereed note.