band -3<=y-x<=4, n=7: |S|, A, B = (1066, 536, 515)
transpose band -4<=y-x<=3, n=7: |S|, A, B = (1066, 536, 515)
square n=3: |S|=6, A=5 (Catalan 5), B=20 (C(2n,n)=20)
square n=4: |S|=24, A=14 (Catalan 14), B=70 (C(2n,n)=70)
square n=5: |S|=120, A=42 (Catalan 42), B=252 (C(2n,n)=252)
worked example: sigma in S: True | sigma in A: True | printed colouring in B: True
