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

1,5

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.

Formula

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

Example

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, ...
  ...

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 *)

Crossrefs

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

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

Main diagonal gives A284645.

Sequence in context: A154221 A026736 A230859 * A050446 A214868 A144048

Adjacent sequences: A213083 A213084 A213085 * A213087 A213088 A213089

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 05 2012

Extensions

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

Status

approved