A213086 Square array read by antidiagonals: T(n,m) (n>=1, m>=0) is the number of partitions of mn that are the sum of m not necessarily distinct partitions of n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 4, 1, 1, 7, 14, 10, 5, 1, 1, 11, 25, 30, 15, 6, 1, 1, 15, 53, 65, 55, 21, 7, 1, 1, 22, 89, 173, 140, 91, 28, 8, 1, 1, 30, 167, 343, 448, 266, 140, 36, 9, 1, 1, 42, 278, 778, 1022, 994, 462, 204, 45, 10, 1, 1, 56, 480, 1518, 2710, 2562, 1974, 750, 285, 55, 11, 1

Offset: 1

Author

N. J. A. Sloane, Jun 05 2012

Keywords

• nonn
• tabl

Examples

			The array begins:
  1,  1,   1,   1,    1,    1,     1,     1,     1,      1, ...
  1,  2,   3,   4,    5,    6,     7,     8,     9,     10, ...
  1,  3,   6,  10,   15,   21,    28,    36,    45,     55, ...
  1,  5,  14,  30,   55,   91,   140,   204,   285,    385, ...
  1,  7,  25,  65,  140,  266,   462,   750,  1155,   1705, ...
  1, 11,  53, 173,  448,  994,  1974,  3606,  6171,  10021, ...
  1, 15,  89, 343, 1022, 2562,  5670, 11418, 21351,  37609, ...
  1, 22, 167, 778, 2710, 7764, 19314, 43164, 88671, 170170, ...
  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..18, flattened
• N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.

Crossrefs

Columns are A000041, A002219, A002220, A002221, A002222. Cf. A213074.

Rows are A000027, A000217, A000330, A001296, A207361.

Main diagonal gives A284645.

Programs

Maple

with(combinat):
g:= proc(n, m) option remember;
      `if`(m>1, map(x-> map(y-> sort([x[], y[]]), g(n, 1))[],
       g(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x),
       {partition(n)[]}), {[]}))
    end:
T:= (n, m)-> nops(g(n, m)):
seq(seq(T(d-m, m), m=0..d-1), d=1..12);  # Alois P. Heinz, Jul 11 2012

Mathematica

T[n_, m_] := Module[{ip, lg, i}, ip = IntegerPartitions[n]; lg = Length[ ip]; i[0]=1; Table[Join[Sequence @@ Table[ip[[i[k]]], {k, 1, m}]] // Sort, Evaluate[Sequence @@ Table[{i[k], i[k-1], lg}, {k, 1, m}]]] // Flatten[#, m-1]& // Union // Length]; T[_, 0] = 1;
Table[T[n-m, m], {n, 1, 12}, {m, 0, n - 1}] // Flatten (* Jean-François Alcover, May 25 2016 *)

Formula

Row n is a polynomial in m: see A213074 for the coefficients.

Extensions

More terms and cross-references from Alois P. Heinz, Jul 11 2012