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Search: id:A121581
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| A121581 |
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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the second column (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. |
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+0
3
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| 1, 1, 1, 1, 3, 2, 1, 9, 11, 3, 1, 33, 43, 39, 4, 1, 153, 193, 199, 169, 5, 1, 873, 1057, 1099, 1081, 923, 6, 1, 5913, 6937, 7147, 7171, 7027, 6117, 7, 1, 46233, 53017, 54187, 54403, 54307, 53413, 47311, 8, 1, 409113, 461257, 468907, 470203, 470323, 469483, 463399
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OFFSET
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1,5
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COMMENT
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Row sums are the factorials (A000142). T(n,0)=1; Sum(k*T(n,k), k=0..n)=A121582
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REFERENCES
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E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
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FORMULA
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The generating polynomial of row n is P(n,s)=Q(n,1,s), where Q(1,t,s)=t and Q(n,t,s)=tQ(n-1,t,s)+(t^n-t)Q(n-1,s,1)/(t-1) for n>=2.
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EXAMPLE
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T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 1 cells in their second columns.
Triangle starts:
1;
1,1;
1,3,2;
1,9,11,3;
1,33,43,39,4;
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MAPLE
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Q[1]:=t: for n from 2 to 11 do Q[n]:=expand(simplify(t*Q[n-1]+(t^n-t)/(t-1)*subs({t=s, s=1}, Q[n-1]))): P[1]:=1: P[n]:=subs(t=1, Q[n]): od: for n from 1 to 11 do seq(coeff(P[n], s, j), j=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000142, A121582, A100822, A121583.
Sequence in context: A126074 A108916 A119421 this_sequence A162976 A106338 A129964
Adjacent sequences: A121578 A121579 A121580 this_sequence A121582 A121583 A121584
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 11 2006
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