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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.
11
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
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
Sequence in context: A181266 A302192 A087060 * A173122 A232723 A275486
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Oct 01 2014
STATUS
approved