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

Author

Alois P. Heinz, Jan 09 2012

Keywords

• nonn
• tabl

Comments

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.

Examples

			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, ...
		

Links

• Wikipedia, Partition of a set

Crossrefs

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.

Programs

Maple

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);

Mathematica

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 *)