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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 11 15:18 EDT 2024. Contains 375073 sequences. (Running on oeis4.)