A121634 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns starting at level 0 (n >= 1; 0 <= k <= n-1).

1, 1, 1, 2, 3, 1, 8, 10, 5, 1, 42, 44, 25, 8, 1, 264, 242, 144, 57, 12, 1, 1920, 1594, 962, 429, 117, 17, 1, 15840, 12204, 7366, 3536, 1131, 219, 23, 1, 146160, 106308, 63766, 32118, 11453, 2664, 380, 30, 1, 1491840, 1036944, 616436, 320710, 123742, 32765, 5704, 620, 38, 1

Offset: 1

Author

Emeric Deutsch, Aug 13 2006

Keywords

• nonn
• tabl

Comments

A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

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 columns with exactly 2 cells starting at level 0.
Triangle starts:
   1;
   1,  1;
   2,  3,  1;
   8, 10,  5,  1;
  42, 44, 25,  8,  1;
		

Links

• E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Crossrefs

Cf. A000142, A121635, A121636, A121585.

Programs

Maple

P[1]:=1: P[2]:=1+t: for n from 3 to 11 do P[n]:=sort(expand((t+n-2)*((n-2)!+P[n-1]))) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form

Mathematica

P[n_ /; n >= 3, t_] := P[n, t] = (t + n - 2) ((n - 2)! + P[n - 1, t]);
P[1, ] = 1; P[2, t] = 1 + t;
Table[CoefficientList[P[n, t], t], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 15 2019 *)

Formula

Row sums are the factorials (A000142).

T(n,0) = A121635(n).

Sum_{k=0..n-1} k*T(n,k) = A121636(n).

The row generating polynomials satisfy P(n,t) = (t+n-2)[(n-2)!+P(n-1,t)] for n >= 3, P(1,t)=1 and P(2,t)=1+t.