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Search: id:A121637
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| A121637 |
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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 2-cell columns (n>=1; 0<=k<=n-1). 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, 2, 3, 1, 7, 10, 6, 1, 29, 47, 33, 10, 1, 147, 265, 210, 82, 15, 1, 889, 1740, 1521, 697, 171, 21, 1, 6252, 13087, 12373, 6377, 1885, 317, 28, 1, 50163, 111066, 112016, 63261, 21390, 4407, 540, 36, 1, 452356, 1050608, 1118991, 680541, 255245, 60903
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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Row sums are the factorials (A000142). T(n,0)=A121638(n). Sum(k*T(n,k), k=0..n-1)=A121639(n)
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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 row generating polynomials are P(n,t)=Q(n,t,1,1), where Q(1,t,x,y)=x, Q(2,t,x,y)=x+ty, and Q(n,t,x,y)=Q(n-1,t,ty,1/t)+(x+ty+n-3)Q(n-1,t,1,1) for n>=3.
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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 horizontal and vertical dominoes, having, respectively, 0 and 1 2-cell columns.
Triangle starts:
1;
1,1;
2,3,1;
7,10,6,1;
29,47,33,10,1;
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MAPLE
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Q[1]:=x: Q[2]:=x+t*y: for n from 3 to 11 do Q[n]:=sort(expand(subs({x=t*y, y=1/t}, Q[n-1])+(x+t*y+n-3)*subs({x=1, y=1}, Q[n-1]))) od: for n from 1 to 11 do P[n]:=sort(subs({x=1, y=1}, Q[n])) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000142, A121638, A121639, A121554.
Adjacent sequences: A121634 A121635 A121636 this_sequence A121638 A121639 A121640
Sequence in context: A114581 A085588 A118008 this_sequence A101175 A050512 A107102
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 13 2006
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