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

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

Offset

1,5

Comments

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).

Links

Table of n, a(n) for n=1..66.

G. Castiglione, A. Frosini, A. Restivo and S. Rinaldi, Enumeration of L-convex polyominoes by rows and columns, Theor. Comp. Sci., 347, 2005, 336-352.

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European Journal of Combinatorics, 28, 2007, 1724-1741 (see Section 3.1.2).

G. Castiglione and A. Restivo, Reconstruction of L-convex polyominoes, Electronic Notes in Discrete Mathematics, Vol. 12, Elsevier Science, 2003.

Formula

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.

T(n,n) = A126765(n-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).

Example

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, ...;

Maple

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

Mathematica

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]}]];
Table[T[m - n + 1, n], {m, 1, 11}, {n, 1, m}] // Flatten (* Jean-François Alcover, Aug 22 2024, after Maple program *)

Crossrefs

Cf. A126765.

Sequence in context: A173043 A082046 A132787 * A119307 A296039 A296974

Adjacent sequences: A181367 A181368 A181369 * A181371 A181372 A181373

Keyword

nonn,tabl

Author

Emeric Deutsch, Oct 18 2010

Status

approved