A121632 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n such that the bottom of the last column is at level k (n>=1; k>=0). 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, 2, 5, 1, 16, 7, 1, 65, 43, 11, 1, 326, 279, 98, 16, 1, 1957, 1999, 867, 194, 22, 1, 13700, 15949, 8068, 2225, 348, 29, 1, 109601, 141291, 80493, 25868, 5009, 580, 37, 1, 986410, 1381219, 865728, 313305, 70949, 10229, 913, 46, 1, 9864101, 14798599

Offset: 1

Author

Emeric Deutsch, Aug 12 2006

Keywords

• nonn
• tabf

Comments

Row n has n-1 terms (n>=2). Row sums are the factorials (A000142). T(n,0)=A000522(n-1). Sum(k*T(n,k), k>=0)=A121633(n).

T(n,k)=number of permutations of {1,2,...,n} that have k left-to-right maxima not in the initial string of consecutive left-to-right maxima. Example: T(4,1)=7 because we have (13)24, (3)124, (3)142, (2)143, (23)14, (3)214 and (3)241; in each of these permutations 4 is the only left-to-right maximum not in the initial string of left-to-right maxima (shown between parentheses). T(4,2)=1 because we have 2134. - Emeric Deutsch, Apr 04 2008

Examples

			T(2,0)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes; the last column of each starts at level 0.
Triangle starts:
1;
2;
5,1;
16,7,1;
65,43,11,1;
326,279,98,16,1;
		

References

• 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, A121633.

Programs

Maple

P[1]:=1: for n from 2 to 12 do P[n]:=sort(expand(1+t*(P[n-1]-1)+(n-1)*P[n-1])) od: 1; for n from 2 to 11 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form

Formula

The row generating polynomials satisfy P(n,t)=1-t+(t+n-1)P(n-1,t) for n>=2 with P(1,t)=1. T(n,k)=T(n-1,k-1)+(n-1)T(n-1,k) for k>=2.