A324009 - OEIS

%I #33 May 11 2024 09:32:19

%S 1,1,1,1,5,1,1,13,13,1,1,25,68,25,1,1,41,222,222,41,1,1,61,555,1110,

%T 555,61,1,1,85,1171,3951,3951,1171,85,1,1,113,2198,11263,19010,11263,

%U 2198,113,1,1,145,3788,27468,70438,70438,27468,3788,145,1

%N The number of convex polyominoes whose smallest bounding rectangle has size w*h (w > 0, h > 0). The table is read by antidiagonals.

%H Mireille Bousquet-Mélou, <a href="https://doi.org/10.1088/0305-4470/25/7/032">Convex polyominoes and algebraic languages</a>, Journal of Physics A25 (1992), 1935-1944.

%H M.-P. Delest and G. Viennot, <a href="http://dx.doi.org/10.1016/0304-3975(84)90116-6">Algebraic languages and polyominoes enumeration</a>, Theoretical Computer Sci., 34 (1984), 169-206.

%H Ira M. Gessel, <a href="http://www.labmath.uqam.ca/~annales/volumes/24-1/PDF/063-066.pdf"> On the number of convex polyominoes</a>, Ann. Sci. Math. Québec 24 (2000), no. 1, 63-66.

%H K. Y. Lin and S. J. Chang, <a href="https://doi.org/10.1088/0305-4470/21/11/020">Rigorous results for the number of convex polygons on the square and honeycomb lattices</a>, Journal of Physics A21 (1988), 2635-2642.

%F a(w, h) = binomial(2w+2h-4, 2w-2) + ((2w+2h-5)/2)*binomial(2w+2h-6, 2w-3) - 2(w+h-3)*binomial(w+h-2, w-1)*binomial(w+h-4, w-2), for w > 0, h > 0.

%F a(w, h) = A093118(w-1, h-1).

%e For w=3 and h=2, the a(3,2)=13 polyominoes spanning a 3 X 2 rectangle are

%e    XXX   X   XX   X    XX

%e    XXX  XXX   XX  XXX  XXX

%e plus all different horizontal and vertical reflections (1+2+2+4+4=13).

%e Table begins

%e 1  1   1    1   1  1 1 ...

%e 1  5  13   25  41 61 ...

%e 1 13  68  222 555 ...

%e 1 25 222 1110 ...

%e 1 41 555 ...

%e 1 61 ...

%e 1 ...

%t Table[Binomial[2 # + 2 h - 4, 2 # - 2] + ((2 # + 2 h - 5)/2) Binomial[2 # + 2 h - 6, 2 # - 3] - 2 (# + h - 3) Binomial[# + h - 2, # - 1] Binomial[# + h - 4, # - 2] &[w - h + 1], {w, 10}, {h, w}] // Flatten (* _Michael De Vlieger_, Apr 15 2019 *)

%o (Sage)

%o def a(w,h):

%o      s = w+h # half the perimeter

%o      return ( binomial(2*s-4,2*w-2) + binomial(2*s-6,2*w-3)*(s-5/2)

%o       - 2*(s-3)*binomial(s-2,w-1)*binomial(s-4,w-2) )

%Y A093118 contains the same data in a different arrangement but without the entries for w=1 and for h=1.

%Y Row sums are A005436.

%K nonn,easy,tabl,walk

%O 1,5

%A _Günter Rote_, Feb 12 2019