A140709 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n in which the maximal number of initial consecutive columns ending at the same level is k (1 <= k <= n).
1, 1, 1, 3, 2, 1, 15, 5, 3, 1, 87, 20, 8, 4, 1, 567, 107, 28, 12, 5, 1, 4167, 674, 135, 40, 17, 6, 1, 34407, 4841, 809, 175, 57, 23, 7, 1, 316647, 39248, 5650, 984, 232, 80, 30, 8, 1, 3219687, 355895, 44898, 6634, 1216, 312, 110, 38, 9, 1, 35878887, 3575582, 400793, 51532, 7850, 1528, 422, 148, 47, 10, 1
Offset: 1
Examples
T(2,1)=1 (the vertical domino); T(2,2)=1 (the horizontal domino); T(3,1)=3 because we have (3), (1,2) and (2,1,1), where (a,b,c,...) stands for a polyomino with columns of lengths a,b,c,..., starting at level 0. Triangle starts: 1; 1, 1; 3, 2, 1; 15, 5, 3, 1; 87, 20, 8, 4, 1; 567, 107, 28, 12, 5, 1;
Links
- G. C. Greubel, Rows n = 1..50 of the triangle, flattened
- 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
T:=proc(n,k) options operator, arrow: binomial(n-1, k-1)+sum(factorial(j)*(j-1)*binomial(n-1-j, k-1),j=2..n-1) end proc: for n to 11 do seq(T(n, k),k=1..n) end do; # yields sequence in triangular form
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Mathematica
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==1, n! -Sum[j!, {j,n-1}], T[n-1, k] + T[n-1, k-1] ]]; Table[T[n, k], {n,14}, {k,n}]//Flatten (* G. C. Greubel, May 02 2021 *)
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PARI
T(n,k) = binomial(n-1, k-1) + sum(j=2, n-1, j!*(j-1)*binomial(n-1-j, k-1)); tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Nov 16 2019
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Sage
@CachedFunction def T(n, k): if (k < 0 or k > n): return 0 elif (k==1): return factorial(n) - sum(factorial(j) for j in (1..n-1)) else: return T(n-1, k-1) + T(n-1, k) flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, May 02 2021
Comments