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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).
4
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.
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
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jan 06 2005, Aug 09 2006
STATUS
approved