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%I #34 Nov 11 2019 21:36:29
%S 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,3,3,3,1,1,1,1,3,4,4,3,
%T 1,1,1,1,4,5,7,5,4,1,1,1,1,4,7,9,9,7,4,1,1,1,1,5,8,14,11,14,8,5,1,1,1,
%U 1,5,10,17,20,20,17,10,5,1,1
%N Number A(n,k) of different ways to divide an n X k rectangle into subsquares, considering only the list of parts; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.
%H Alois P. Heinz and Christopher Hunt Gribble, <a href="/A224697/b224697.txt">Antidiagonals n = 0..27, flattened</a> (first 25 antidiagonals from Alois P. Heinz)
%e A(4,5) = 9 because there are 9 ways to divide a 4 X 5 rectangle into subsquares, considering only the list of parts: [20(1 X 1)], [16(1 X 1), 1(2 X 2)], [12(1 X 1), 2(2 X 2)], [11(1 X 1), 1(3 X 3)], [8(1 X 1), 3(2 X 2)], [7(1 X 1), 1(2 X 2), 1(3 X 3)], [4(1 X 1), 4(2 X 2)], [4(1 X 1), 1(4 X 4)], [3(1 X 1), 2(2 X 2), 1(3 X 3)]. There is no way to divide this rectangle into [2(1 X 1), 2(3 X 3)].
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...
%e 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, ...
%e 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, ...
%e 1, 1, 3, 5, 9, 11, 20, 26, 36, 48, ...
%e 1, 1, 4, 7, 14, 20, 31, 47, 71, 95, ...
%e 1, 1, 4, 8, 17, 26, 47, 57, 102, 143, ...
%e 1, 1, 5, 10, 24, 36, 71, 102, 148, 238, ...
%e 1, 1, 5, 12, 29, 48, 95, 143, 238, 312, ...
%p b:= proc(n, l) option remember; local i, k, s, t;
%p if max(l[])>n then {} elif n=0 or l=[] then {[]}
%p elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
%p else for k do if l[k]=0 then break fi od; s:={};
%p for i from k to nops(l) while l[i]=0 do s:=s union
%p map(x->sort([x[], 1+i-k]), b(n, [l[j]$j=1..k-1,
%p 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
%p od; s
%p fi
%p end:
%p A:= (n, k)-> `if`(n>=k, nops(b(n, [0$k])), nops(b(k, [0$n]))):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t b[n_, l_] := b[n, l] = Module[{i, k, m, s, t}, Which[Max[l] > n, {}, n == 0 || l == {}, {{}}, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[Function[x, Sort[Append[x, 1+i-k]]], b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]] ] ]]]; s]]; a[n_, k_] := If[n >= k, Length @ b[n, Array[0&, k]], Length @ b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 19 2013, translated from Maple *)
%Y Columns (or rows) k=0+1, 2-5 give: A000012, A008619, A001399, A008763(n+4), A187753.
%Y Main diagonal gives: A034295.
%Y Cf. A225622.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Apr 15 2013
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