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A230505 T(n,k,s) is the number of parts of each size in the set of partitions of an n X k rectangle into integer-sided squares with side s, considering only the list of parts; irregular triangle T(n,k,s), n >= k >= s >= 1, read by rows.

%I A230505 #42 Oct 27 2023 22:06:07
%S A230505 1,2,4,1,3,8,1,14,1,1,4,12,3,27,3,1,47,10,1,1,5,18,3,41,4,2,85,13,3,1,
%T A230505 134,16,4,1,1,6,24,6,62,7,4,135,27,5,3,250,40,13,3,1,415,82,24,6,1,1,
%U A230505 7,32,6,87,9,5,204,34,8,4,381,53,18,5,3,717,127,45,13,4,1
%N A230505 T(n,k,s) is the number of parts of each size in the set of partitions of an n X k rectangle into integer-sided squares with side s, considering only the list of parts; irregular triangle T(n,k,s), n >= k >= s >= 1, read by rows.
%H A230505 Alois P. Heinz, <a href="/A230505/b230505.txt">Rows n = 1..78, flattened</a> (Rows 1..42 from Christopher Hunt Gribble)
%H A230505 Christopher Hunt Gribble, <a href="/A230505/a230505_1.cpp.txt">C++ program</a>
%F A230505 Sum_{s=1..k} T(n,k,s) = A225622(n,k).
%F A230505 Sum_{s=1..k} T(n,k,s)*s^2 = n*k*A224697(n,k).
%e A230505 T(5,4,2) = 13 because there are 13 2 X 2 squares in the 9 partitions of a 5 X 4 rectangle into integer-sided squares.  The partitions are:
%e A230505 .         Square side
%e A230505 .         1  2  3  4
%e A230505 1        20  0  0  0
%e A230505 2        16  1  0  0
%e A230505 3        12  2  0  0
%e A230505 4         8  3  0  0
%e A230505 5         4  4  0  0
%e A230505 6        11  0  1  0
%e A230505 7         7  1  1  0
%e A230505 8         3  2  1  0
%e A230505 9         4  0  0  1
%e A230505 Total    85 13  3  1
%e A230505 The irregular triangle begins:
%e A230505 n,k          Square Side (s)
%e A230505 .       1   2   3   4   5   6   7 ...
%e A230505 1,1     1
%e A230505 2,1     2
%e A230505 2,2     4   1
%e A230505 3,1     3
%e A230505 3,2     8   1
%e A230505 3,3    14   1   1
%e A230505 4,1     4
%e A230505 4,2    12   3
%e A230505 4,3    27   3   1
%e A230505 4,4    47  10   1   1
%e A230505 5,1     5
%e A230505 5,2    18   3
%e A230505 5,3    41   4   2
%e A230505 5,4    85  13   3   1
%e A230505 5,5   134  16   4   1   1
%e A230505 6,1     6
%e A230505 6,2    24   6
%e A230505 6,3    62   7   4
%e A230505 6,4   135  27   5   3
%e A230505 6,5   250  40  13   3   1
%e A230505 6,6   415  82  24   6   1   1
%e A230505 7,1     7
%e A230505 7,2    32   6
%e A230505 7,3    87   9   5
%e A230505 7,4   204  34   8   4
%e A230505 7,5   381  53  18   5   3
%e A230505 7,6   717 127  45  13   4   1
%e A230505 7,7  1102 165  60  16   6   1   1
%p A230505 b:= proc(n, l) option remember; local i, k, s, t;
%p A230505       if max(l[])>n then {} elif n=0 or l=[] then {0}
%p A230505     elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
%p A230505     else for k do if l[k]=0 then break fi od; s:={};
%p A230505          for i from k to nops(l) while l[i]=0 do s:=s union
%p A230505              map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,
%p A230505                  1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
%p A230505          od; s
%p A230505       fi
%p A230505     end:
%p A230505 T:= (n, k)->(p->seq(coeff(p, x, v), v=1..k))(add(h, h=b(n, [0$k]))):
%p A230505 seq(seq(T(n, k), k=1..n), n=1..9);  # _Alois P. Heinz_, Oct 24 2013
%t A230505 b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, {}, n == 0 || l == {}, {0}, Min[l] > 0, t = Min[l]; b[n - t, l - t], True, For[k = 1, k <= Length[l], k++, If[l[[k]] == 0, Break[]]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[# + x^(1 + i - k)&, b[n, Join[l[[1 ;; k - 1]], Array[1 + i - k&, i - k + 1], l[[i + 1 ;; Length[l]]]]]]]; s]]; T[n_, k_] := Function[p, Table[Coefficient[p, x, v], {v, 1, k}]][b[n, Array[0&, k]] // Total]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 9}] // Flatten (* _Jean-François Alcover_, Jan 24 2016, after _Alois P. Heinz_ *)
%Y A230505 Cf. A034295, A224697, A225622.
%K A230505 nonn,tabf
%O A230505 1,2
%A A230505 _Christopher Hunt Gribble_, Oct 22 2013