A121579 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having along the lower contour exactly k reentrant corners, i.e., a vertical step that is followed by a horizontal step (n>=1, k>=0).

1, 2, 5, 1, 16, 8, 65, 52, 3, 326, 344, 50, 1957, 2473, 595, 15, 13700, 19676, 6524, 420, 109601, 173472, 71862, 7840, 105, 986410, 1686912, 823836, 127232, 4410, 9864101, 17981193, 9976686, 1975750, 118125, 945, 108505112, 208769296, 128350992

Offset: 1

Author

Emeric Deutsch, Aug 08 2006

Keywords

• nonn
• tabf

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.

Row n contains ceiling(n/2) terms.

Row sums are the factorials (A000142).

T(n,0) = A000522(n).

T(2n+1,n) = (2n-1)!! = A001147(n) (the double factorials).

Sum_{k=0..n} k*T(n,k) = A002538(n-2) for n >= 3.

Examples

			T(2,0)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having no reentrant corners along the lower contour.
Triangle starts:
    1;
    2;
    5,   1;
   16,   8;
   65,  52,   3;
  326, 344,  50;
		

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, A000522, A001147, A002538.

Programs

Maple

Q[1]:=1: for n from 2 to 13 do Q[n]:=sort(expand(subs(x=t,Q[n-1])+(n-1)*x*subs(x=1,Q[n-1]))) od: for n from 1 to 13 do P[n]:=subs(x=1,Q[n]) od: for n from 1 to 13 do seq(coeff(P[n],t,j),j=0..ceil(n/2)-1) od; # yields sequence in triangular form

Formula

The row generating polynomials are P(n,t) = Q(n,t,1), where Q(1,t,x) = 1 and Q(n,t,x) = Q(n-1,t,t) + (n-1)xQ(n-1,t,1) for n >= 2.