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A181370
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Square array read by antidiagonals: T(m,n) is the number of L-convex polyominoes with m rows and n columns.
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0
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1, 1, 1, 1, 5, 1, 1, 11, 11, 1, 1, 19, 42, 19, 1, 1, 29, 110, 110, 29, 1, 1, 41, 235, 402, 235, 41, 1, 1, 55, 441, 1135, 1135, 441, 55, 1, 1, 71, 756, 2709, 4070, 2709, 756, 71, 1, 1, 89, 1212, 5740, 11982, 11982, 5740, 1212, 89, 1, 1, 109, 1845, 11124, 30618, 42510, 30618, 11124, 1845, 109, 1
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OFFSET
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1,5
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COMMENTS
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An L-convex polyomino is a convex polyomino where any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientation of the letter L).
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LINKS
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FORMULA
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T(m,n) = Sum(2^k*(k^2+(m+n-2)*k+mn-1)*binomial(m+n-2,2k)*binomial(m+n-2-2k,m-1-k)/[(m+n-2)(2k+1)], k=0..min(m-1,n-1)) ((m,n)!=(1,1)); T(1,1)=1.
G.f.: G(x,y) = Sum_{m>=1, n>=1} T(m,n)*x^m*y^n = xy(1-x)(1-y)/(1-2x-2y+x^2+y^2) (see 2nd Maple program).
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EXAMPLE
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T(2,2)=5 because we have the 2 X 2 square and the four polyominoes obtained by removing 1 square from the four squares of the 2 X 2 square.
Square array starts:
1, 1, 1, 1, 1, ...;
1, 5, 11, 19, 29, ...;
1, 11, 42, 110, 235, ...;
1, 19, 110, 402, 1135, ...;
1, 29, 235, 1135, 4070, ...;
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MAPLE
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T := proc (m, n) if m = 1 and n = 1 then 1 else sum(2^k*(k^2+(m+n-2)*k+m*n-1)*binomial(m+n-2, 2*k)*binomial(m+n-2-2*k, m-1-k)/((m+n-2)*(2*k+1)), k = 0 .. min(m-1, n-1)) end if end proc: matrix(9, 9, T); # yields first 9 rows and 9 columns of the square array
G := x*y*(1-x)*(1-y)/(1-2*x-2*y+x^2+y^2): a := proc (m, n) options operator, arrow; coeff(series(coeff(simplify(series(G, x = 0, 20)), x, m), y = 0, 20), y, n) end proc: matrix(9, 9, a); # yields first 9 rows and 9 columns of the square array
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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