A100822 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the first column. (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, 6, 8, 9, 1, 24, 30, 32, 33, 1, 120, 144, 150, 152, 153, 1, 720, 840, 864, 870, 872, 873, 1, 5040, 5760, 5880, 5904, 5910, 5912, 5913, 1, 40320, 45360, 46080, 46200, 46224, 46230, 46232, 46233, 1, 362880, 403200, 408240, 408960, 409080, 409104, 409110, 409112, 409113, 1

Offset

1,4

Comments

Row n has n terms. Rows are circular permutations of the rows of A054115. Column 1 and row sums yield A000142 (the factorial numbers). Column 2 yields A059171.

T(n+1,n)=A007489(n).

References

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

Links

Table of n, a(n) for n=1..55.

Formula

T(n, k)=sum((n-j)!, j=1..k) for 1<=k<n; T(n, n)=1.

T(n,k)=T(n-1,k-1)+(n-1)! for k<n; T(n,n)=1.

Example

Triangle begins:
1;
1,1;
2,3,1;
6,8,9,1;
24,30,32,33,1;
T(2,1)=T(2,2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 1 and 2 cells in their first columns.

Maple

T:=proc(n, k) if k=n then 1 elif k<n then sum((n-j)!, j=1..k) else 0 fi end: for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

Crossrefs

Cf. A000142, A054115, A059171, A007489.

Sequence in context: A279038 A092271 A054115 * A347766 A198427 A289905

Adjacent sequences: A100819 A100820 A100821 * A100823 A100824 A100825

Keyword

nonn,tabl

Author

Emeric Deutsch, Jan 06 2005, Aug 09 2006

Status

approved