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

1,2

Links

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

Example

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;

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 *)

Crossrefs

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

Sequence in context: A181266 A302192 A087060 * A173122 A232723 A275486

Adjacent sequences: A248109 A248110 A248111 * A248113 A248114 A248115

Keyword

nonn,tabf

Author

Alois P. Heinz, Oct 01 2014

Status

approved