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Search: id:A140709
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| A140709 |
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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n in which the maximal number of initial consecutive columns ending at the same level is k (1<=k<=n). (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
2
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| 1, 1, 1, 3, 2, 1, 15, 5, 3, 1, 87, 20, 8, 4, 1, 567, 107, 28, 12, 5, 1, 4167, 674, 135, 40, 17, 6, 1, 34407, 4841, 809, 175, 57, 23, 7, 1, 316647, 39248, 5650, 984, 232, 80, 30, 8, 1, 3219687, 355895, 44898, 6634, 1216, 312, 110, 38, 9, 1, 35878887, 3575582, 400793
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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Sum of entries in row n is n! (A000142).
T(n,1)=A132371(n).
Sum(k*T(n,k),k=1..n)=A140710(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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T(n,k)=binom(n-1,k-1)+sum(j!*(j-1)*binom(n-1-j,k-1),j=2..n-1). Rec. rel.: T(n,k)=T(n-1,k)+T(n-1,k-1) for n,k>=2.
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EXAMPLE
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T(2,1)=1 (the vertical domino); T(2,2)=1 (the horizontal domino); T(3,1)=3 because we have (3), (1,2), and (2,1,1), where (a,b,c,...) stands for a polyomino with columns of lengths a,b,c,..., starting at level 0.
Triangle starts:
1;
1,1;
3,2,1;
15,5,3,1;
87,20,8,4,1;
567,107,28,12,5,1;
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MAPLE
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T:=proc(n, k) options operator, arrow: binomial(n-1, k-1)+sum(factorial(j)*(j-1)*binomial(n-1-j, k-1), j=2..n-1) end proc: for n to 11 do seq(T(n, k), k=1..n) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A000142, A132371, A140710.
Sequence in context: A127126 A112911 A111548 this_sequence A109282 A135902 A135876
Adjacent sequences: A140706 A140707 A140708 this_sequence A140710 A140711 A140712
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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), Jun 03 2008
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