A203986
Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of {1,2,...,k*n} into n k-element subsets having the same sum.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 4, 2, 1, 0, 1, 1, 1, 0, 32, 0, 1, 0, 1, 1, 1, 29, 305, 392, 11, 1, 0, 1, 1, 1, 0, 4331, 0, 6883, 0, 1, 0, 1, 1, 1, 263, 63261, 2097719, 3245664, 171088, 84, 1, 0, 1, 1, 1, 0, 1025113, 0, 2549091482, 0, 5661874, 0, 1, 0, 1
Offset: 0
A(0,0) = 1.
A(1,1) = 1: {1}.
A(2,2) = 1: {1,4}, {2,3}.
A(3,3) = 2: {1,5,9}, {2,6,7}, {3,4,8}; {1,6,8}, {2,4,9}, {3,5,7}.
A(4,2) = 1: {1,8}, {2,7}, {3,6}, {4,5}.
A(2,4) = 4: {1,2,7,8}, {3,4,5,6}; {1,3,6,8}, {2,4,5,7}; {1,4,5,8}, {2,3,6,7}; {1,4,6,7}, {2,3,5,8}.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 0, 1, 0, 4, 0, 29, ...
1, 0, 1, 2, 32, 305, 4331, ...
1, 0, 1, 0, 392, 0, 2097719, ...
1, 0, 1, 11, 6883, 3245664, 2549091482, ...
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b:= proc() option remember; local i, j, t, k, m; m:= args[nargs-1]; k:= args[nargs]; if args[1]=0 then `if`(nargs=3, 1, b(args[t] $t=2..nargs)) elif args[1]<1 then 0 else add(`if`(args[j] `if`(n=0 or k=0, 1, b((k*(n*k+1)/2 +k/97) $n, k*n, k)/n!):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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b[args_List] := b[args] = Module[{nargs = Length[args], k = args[[-1]], m = args[[-2]]}, Which[args[[1]] == 0, If[nargs == 3, 1, b[args[[2 ;; nargs]]]], args[[1]] < 1, 0, True, Sum[If[args[[j]] < m, 0, b[Join[Sort[Table[args[[i]] - If[i == j, m + 1/97, 0], {i, 1, nargs - 2}]], {m - 1, k}]]], {j, 1, nargs - 2}] ]]; A[n_, k_] := If[n == 0 || k == 0, 1, b[Join[Array[(k*(n*k + 1)/2 + k/97) &, n], {k*n, k}]]/n!]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)
A264813
Number of permutations of 3 indistinguishable copies of 1,...,n such that the first and second copies of j are adjacent and there are exactly j numbers between the second and the third copy of j.
Original entry on oeis.org
1, 0, 1, 1, 0, 3, 6, 0, 53, 199, 0, 2908, 13699, 0, 369985, 2135430, 0, 87265700, 611286653, 0
Offset: 0
a(0) = 1: the empty permutation.
a(2) = 1: 221121.
a(3) = 1: 223321131.
a(5) = 3: 223325534411514, 225523344531141, 552244253341131.
a(6) = 6: 221121665544336543, 225523366534411614, 225526633544361141, 446611415563322532, 552266253344631141, 665544336543221121.
A203017
Number of partitions of {1,2,...,3n} into 3 n-element subsets having the same sum.
Original entry on oeis.org
1, 0, 1, 2, 32, 305, 4331, 63261, 1025113, 17495345, 313692810, 5838204047, 112185853894, 2213711510395, 44691175805738, 920173212324164, 19274796589413439, 409908483736507979, 8835309887111026335, 192739853119591626715, 4250191938786946069812, 94641409538083474973850
Offset: 0
a(0) = 1: {}, {}, {}.
a(1) = 0: there is no partition of {1,2,3} into 3 1-element subsets having the same sum.
a(2) = 1: {1,6}, {2,5}, {3,4}.
a(3) = 2: {1,5,9}, {2,6,7}, {3,4,8}; {1,6,8}, {2,4,9}, {3,5,7}.
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b:= proc() option remember; local i, j, t, k, m; m:= args[nargs-1]; k:= args[nargs-0]; if args[1]=0 then `if`(nargs=3, 1, b(args[t]$t=2..nargs)) elif args[1]<1 then 0 else add(`if`(args[j] b((n*(3*n+1)/2 +n/97)$3, 3*n, n)/`if`(n>0, 6, 1):
seq(a(n), n=0..10);
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b[args_] := b[args] = Module[{nargs = Length[args], k = args[[-1]], m = args[[-2]]}, Which[args[[1]] == 0, If[nargs == 3, 1, b[args[[2 ;; nargs]] ]], args[[1]]<1, 0, True, Sum[If[args[[j]] < m, 0, b[Join[Sort[Table[ args[[i]] - If[i == j, m + 1/97, 0], {i, 1, nargs - 2}]], {m - 1, k}]]], {j, 1, nargs-2}]]];
A[n_, k_] := If[n == 0 || k == 0, 1, b[Join[Array[(k*(n*k + 1)/2 + k/97)&, n], {k*n, k}]]/n!];
a[k_] := A[3, k];
a /@ Range[0, 10] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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