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A181370 Square array read by antidiagonals: T(m,n) is the number of L-convex polyominoes with m rows and n columns.

%I A181370 #20 Aug 22 2024 14:42:07
%S A181370 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,
%T A181370 235,41,1,1,55,441,1135,1135,441,55,1,1,71,756,2709,4070,2709,756,71,
%U A181370 1,1,89,1212,5740,11982,11982,5740,1212,89,1,1,109,1845,11124,30618,42510,30618,11124,1845,109,1
%N A181370 Square array read by antidiagonals: T(m,n) is the number of L-convex polyominoes with m rows and n columns.
%C A181370 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).
%H A181370 G. Castiglione, A. Frosini, A. Restivo and S. Rinaldi, <a href="https://doi.org/10.1016/j.tcs.2005.06.031">Enumeration of L-convex polyominoes by rows and columns</a>, Theor. Comp. Sci., 347, 2005, 336-352.
%H A181370 G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, <a href="https://doi.org/10.1016/j.ejc.2006.06.020">Combinatorial aspects of L-convex polyominoes</a>, European Journal of Combinatorics, 28, 2007, 1724-1741 (see Section 3.1.2).
%H A181370 G. Castiglione and A. Restivo, <a href="https://doi.org/10.1016/S1571-0653(04)00494-9">Reconstruction of L-convex polyominoes</a>, Electronic Notes in Discrete Mathematics, Vol. 12, Elsevier Science, 2003.
%F A181370 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.
%F A181370 T(n,n) = A126765(n-1).
%F A181370 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).
%e A181370 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.
%e A181370 Square array starts:
%e A181370   1,  1,   1,    1,    1, ...;
%e A181370   1,  5,  11,   19,   29, ...;
%e A181370   1, 11,  42,  110,  235, ...;
%e A181370   1, 19, 110,  402, 1135, ...;
%e A181370   1, 29, 235, 1135, 4070, ...;
%p A181370 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
%p A181370 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
%t A181370 T[m_, n_] := If[m == 1 && n == 1, 1, 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]}]];
%t A181370 Table[T[m - n + 1, n], {m, 1, 11}, {n, 1, m}] // Flatten (* _Jean-François Alcover_, Aug 22 2024, after Maple program *)
%Y A181370 Cf. A126765.
%K A181370 nonn,tabl
%O A181370 1,5
%A A181370 _Emeric Deutsch_, Oct 18 2010