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A324009 The number of convex polyominoes whose smallest bounding rectangle has size w*h (w > 0, h > 0). The table is read by antidiagonals.

%I A324009 #33 May 11 2024 09:32:19
%S A324009 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 A324009 555,61,1,1,85,1171,3951,3951,1171,85,1,1,113,2198,11263,19010,11263,
%U A324009 2198,113,1,1,145,3788,27468,70438,70438,27468,3788,145,1
%N A324009 The number of convex polyominoes whose smallest bounding rectangle has size w*h (w > 0, h > 0). The table is read by antidiagonals.
%H A324009 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 A324009 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 A324009 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 A324009 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 A324009 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 A324009 a(w, h) = A093118(w-1, h-1).
%e A324009 For w=3 and h=2, the a(3,2)=13 polyominoes spanning a 3 X 2 rectangle are
%e A324009    XXX   X   XX   X    XX
%e A324009    XXX  XXX   XX  XXX  XXX
%e A324009 plus all different horizontal and vertical reflections (1+2+2+4+4=13).
%e A324009 Table begins
%e A324009 1  1   1    1   1  1 1 ...
%e A324009 1  5  13   25  41 61 ...
%e A324009 1 13  68  222 555 ...
%e A324009 1 25 222 1110 ...
%e A324009 1 41 555 ...
%e A324009 1 61 ...
%e A324009 1 ...
%t A324009 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 A324009 (Sage)
%o A324009 def a(w,h):
%o A324009      s = w+h # half the perimeter
%o A324009      return ( binomial(2*s-4,2*w-2) + binomial(2*s-6,2*w-3)*(s-5/2)
%o A324009       - 2*(s-3)*binomial(s-2,w-1)*binomial(s-4,w-2) )
%Y A324009 A093118 contains the same data in a different arrangement but without the entries for w=1 and for h=1.
%Y A324009 Row sums are A005436.
%K A324009 nonn,easy,tabl,walk
%O A324009 1,5
%A A324009 _Günter Rote_, Feb 12 2019