A321230 -id:A321230 - cp's OEIS frontend

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.

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

Alois P. Heinz, Jan 09 2012

A(n,k) = 0 if n>1 and k>0 and (k=1 or k*(n-1) mod 2 = 1).

The element sum of each subset is k*(k*n+1)/2.

			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, ...

Wikipedia, Partition of a set

Cf. A168238 (bisection of row n=2), A203017 (row n=3), A104185 (bisection of column k=3), A203435 (column k=4).

Main diagonal gives A321230.

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);

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 *)

A321282 Number of set partitions of [n^2] into n subsets having the same sum.

Original entry on oeis.org

1, 1, 1, 9, 2650, 100664383, 808087012923418

Offset: 0

Alois P. Heinz, Nov 01 2018

			a(3) = 9: 12345|69|78, 1239|456|78, 1248|357|69, 1257|348|69, 1347|258|69, 1356|249|78, 159|2346|78, 168|249|357, 159|267|348.

Wikipedia, Partition of a set

Cf. A007785, A275714, A317807, A321230.

b:= proc(l, n) option remember; `if`(n=0, 1, add(`if`(n>l[j],
       0, b(sort(subsop(j=l[j]-n, l)), n-1)), j=1..nops(l)))
    end:
a:= n-> b([n*(1+n^2)/2$n], n^2)/n!:
seq(a(n), n=0..5);

b[l_, n_] := b[l, n] = If[n == 0, 1, Sum[If[n > l[[j]], 0, b[Sort[ ReplacePart[l, j -> l[[j]] - n]], n-1]], {j, 1, Length[l]}]];
a[n_] := b[Table[n(1+n^2)/2, {n}], n^2]/n!;
a /@ Range[0, 5] (* Jean-François Alcover, May 04 2020, after Maple *)

a(n) = A275714(n^2,n).

cp's OEIS Frontend

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.

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