A196231 Irregular triangle T(n,k), n>=1, 1<=k<=ceiling(n/2), read by rows: T(n,k) is the number of different ways to select k disjoint (nonempty) subsets from {1..n} with equal element sum.

1, 3, 7, 1, 15, 3, 31, 7, 1, 63, 17, 3, 127, 43, 8, 1, 255, 108, 22, 3, 511, 273, 63, 9, 1, 1023, 708, 157, 23, 3, 2047, 1867, 502, 67, 10, 1, 4095, 4955, 1562, 203, 26, 3, 8191, 13256, 4688, 693, 83, 11, 1, 16383, 35790, 15533, 2584, 322, 30, 3, 32767, 97340

Offset: 1

Author

Alois P. Heinz, Sep 29 2011

Keywords

• nonn
• tabf

Examples

			T(8,4) = 3: {1,6}, {2,5}, {3,4}, {7} have element sum 7, {1,7}, {2,6}, {3,5}, {8} have element sum 8, and {1,8}, {2,7}, {3,6}, {4,5} have element sum 9.
Triangle begins:
.   1;
.   3;
.   7,   1;
.  15,   3;
.  31,   7,  1;
.  63,  17,  3;
. 127,  43,  8, 1;
. 255, 108, 22, 3;
		

Links

• Alois P. Heinz, Rows n = 1..26, flattened

Crossrefs

Columns k=1-10 give: A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196235, A196236, A196237. Row sums give A196534. Row lengths are in A110654.

Programs

Maple

b:= proc(l, n, k) option remember; local i, j; `if`(l=[0$k], 1, `if`(add(j, j=l)>n*(n-1)/2, 0, b(l, n-1, k))+ add(`if`(l[j] -n<0, 0, b(sort([seq(l[i] -`if`(i=j, n, 0), i=1..k)]), n-1, k)), j=1..k)) end: T:= (n, k)-> add(b([t$k], n, k), t=2*k-1..floor(n*(n+1)/(2*k)))/k!:
seq(seq(T(n, k), k=1..ceil(n/2)), n=1..15);

Mathematica

b[l_List, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0&, k], 1, If [Total[l] > n*(n-1)/2, 0, b[l, n-1, k]] + Sum [If [l[[j]] - n < 0, 0, b[Sort[Table[l[[i]] - If[i == j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k}]] ]; T[n_, k_] := Sum[b[Array[t&, k], n, k], {t, 2*k-1, Floor[n*(n+1)/(2*k)]}]/k!; Table[Table[T[n, k], {k, 1, Ceiling[n/2]}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)