A248112 Number T(n,k) of subsets of {1,...,n} containing n and having at least one set partition into k blocks with equal element sum; triangle T(n,k), n>=1, 1<=k<=floor((n+1)/2), read by rows.

1, 2, 4, 1, 8, 2, 16, 4, 1, 32, 10, 2, 64, 20, 5, 1, 128, 44, 12, 2, 256, 93, 29, 6, 1, 512, 198, 63, 14, 2, 1024, 414, 146, 37, 7, 1, 2048, 864, 329, 88, 16, 2, 4096, 1788, 722, 218, 49, 8, 1, 8192, 3687, 1613, 515, 118, 19, 2, 16384, 7541, 3505, 1226, 313, 62, 9, 1

Offset: 1

Author

Alois P. Heinz, Oct 01 2014

Keywords

• nonn
• tabf

Examples

			T(7,3) = 5: {2,3,4,5,7}-> 25/34/7, {1,3,4,6,7}-> 16/34/7, {1,2,5,6,7}-> 16/25/7, {1,2,3,5,6,7}-> 17/26/35, {2,3,4,5,6,7}-> 27/36/45.
T(8,4) = 2: {1,2,3,5,6,7,8}-> 17/26/35/8, {1,2,3,4,5,6,7,8}-> 18/27/36/45.
T(9,5) = 1: {1,2,3,5,6,7,8,9}-> 18/27/36/45/9.
Triangle T(n,k) begins:
01 :    1;
02 :    2;
03 :    4,   1;
04 :    8,   2;
05 :   16,   4,   1;
06 :   32,  10,   2;
07 :   64,  20,   5,  1;
08 :  128,  44,  12,  2;
09 :  256,  93,  29,  6,  1;
10 :  512, 198,  63, 14,  2;
11 : 1024, 414, 146, 37,  7, 1;
12 : 2048, 864, 329, 88, 16, 2;
		

Links

• Alois P. Heinz, Rows n = 0..24, flattened

Crossrefs

Columns k=1-10 give: A000079(n-1), A232466, A232534, A248113, A248114, A248115, A248116, A248117, A248118, A248119.

Programs

Maple

b:= proc(l, i) option remember; local k, r, j;
      k, r:= nops(l), {};
      if i*(i+1)/2 < l[-1]*k-add(j, j=l) then r
    elif i=0 then {r}
    else for j to k do r:= r union map(y->y union {i}, b((p->
           map(x->x-p[1], p))(sort(subsop(j=l[j]+i, l))), i-1))
         od;
         r union b(l, i-1)
      fi
    end:
A:= (n, k)-> `if`(k=1, 2^(n-1), nops(b([0$(k-1), n], n-1))):
seq(seq(A(n, k), k=1..iquo(n+1, 2)), n=1..15);

Mathematica

b[l_, i_] := b[l, i] = Module[{k, r, j}, {k, r} = {Length[l], {}}; Which[ i*(i+1)/2 < l[[-1]]*k - Total[l], r, i == 0, {r}, True, For[j = 1, j <= k, j++, r = r ~Union~ Map[# ~Union~ {i}&, b[Function[p, Map[#-p[[1]]&, p] ][Sort[ReplacePart[l, j -> l[[j]]+i]]], i-1]]]; r ~Union~ b[l, i-1]]]; A[n_, k_] := If[k==1, 2^(n-1), Length[b[Append[Array[0&, (k-1)], n], n-1] ]]; Table[A[n, k], {n, 1, 15}, {k, 1, Quotient[n+1, 2]}] // Flatten (* Jean-François Alcover, Feb 03 2017, Translated from Maple *)