A121637 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns (n>=1; 0<=k<=n-1). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
1, 1, 1, 2, 3, 1, 7, 10, 6, 1, 29, 47, 33, 10, 1, 147, 265, 210, 82, 15, 1, 889, 1740, 1521, 697, 171, 21, 1, 6252, 13087, 12373, 6377, 1885, 317, 28, 1, 50163, 111066, 112016, 63261, 21390, 4407, 540, 36, 1, 452356, 1050608, 1118991, 680541, 255245, 60903, 9247, 863, 45, 1
Offset: 1
Examples
T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 0 and 1 2-cell columns. Triangle starts: 1; 1, 1; 2, 3, 1; 7, 10, 6, 1; 29, 47, 33, 10, 1; ...
References
- E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
Programs
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Maple
Q[1]:=x: Q[2]:=x+t*y: for n from 3 to 11 do Q[n]:=sort(expand(subs({x=t*y,y=1/t},Q[n-1])+(x+t*y+n-3)*subs({x=1,y=1},Q[n-1]))) od: for n from 1 to 11 do P[n]:=sort(subs({x=1,y=1},Q[n])) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form
Formula
The row generating polynomials are P(n,t)=Q(n,t,1,1), where Q(1,t,x,y)=x, Q(2,t,x,y)=x+ty and Q(n,t,x,y)=Q(n-1,t,ty,1/t)+(x+ty+n-3)Q(n-1,t,1,1) for n>=3.
Comments