A134979 Triangle read by rows: T(n,k) = number of partitions of n where the maximum number of objects in partitions of any given size is k.

1, 0, 2, 0, 1, 2, 0, 1, 1, 3, 0, 0, 3, 2, 2, 0, 0, 2, 4, 1, 4, 0, 0, 1, 6, 3, 3, 2, 0, 0, 1, 6, 4, 6, 1, 4, 0, 0, 0, 6, 7, 8, 3, 3, 3, 0, 0, 0, 5, 7, 14, 4, 6, 2, 4, 0, 0, 0, 5, 7, 18, 7, 9, 5, 3, 2, 0, 0, 0, 3, 10, 22, 9, 14, 6, 6, 1, 6, 0, 0, 0, 2, 9, 26, 15, 19, 11, 9, 3, 5, 2

Offset: 1

Author

Franklin T. Adams-Watters, Feb 04 2008

Keywords

• nonn
• look
• tabl

Comments

Every column is eventually 0; the last row with a nonzero value in column k is A024916(k). T(A024916(k)-i, k) <= P(i), where P is the partition function (A000041); equality holds for 0 <= i <= k. The partition represented by the last number in column k is row k of A010766.

Examples

			For the partition [3,2^2], there are 3 objects in the part of size 3 and 4 objects in the parts of size 2, so this partition is counted towards T(7,4).
Triangle T(n,k) begins:
  1;
  0, 2;
  0, 1, 2;
  0, 1, 1, 3;
  0, 0, 3, 2,  2;
  0, 0, 2, 4,  1,  4;
  0, 0, 1, 6,  3,  3, 2;
  0, 0, 1, 6,  4,  6, 1,  4;
  0, 0, 0, 6,  7,  8, 3,  3, 3;
  0, 0, 0, 5,  7, 14, 4,  6, 2, 4;
  0, 0, 0, 5,  7, 18, 7,  9, 5, 3, 2;
  0, 0, 0, 3, 10, 22, 9, 14, 6, 6, 1, 6;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened

Crossrefs

Cf. A008284, A091602, A000041 (row sums), A000005 (main diagonal), A032741 (2nd diagonal), A010766.

Column sums give A332233.

Programs

Maple

b:= proc(n, i, m) option remember; `if`(n=0 or i=1, x^
      max(m, n), add(b(n-i*j, i-1, max(m, i*j)), j=0..n/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2, 0)):
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 07 2020

Mathematica

b[n_, i_, m_] := b[n, i, m] = If[n == 0 || i == 1, x^Max[m, n], Sum[b[n - i j, i - 1, Max[m, i j]], {j, 0, n/i}]];
T[n_] := Table[Coefficient[b[n, n, 0], x, i], {i, 1, n}];
Array[T, 20] // Flatten (* Jean-François Alcover, Nov 10 2020, after Alois P. Heinz *)