A350647 Number T(n,k) of partitions of [n] having k blocks containing their own index when blocks are ordered with decreasing largest elements; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows.

1, 0, 1, 1, 1, 1, 3, 1, 6, 7, 2, 16, 25, 10, 1, 73, 91, 35, 4, 298, 390, 163, 25, 1, 1453, 1797, 755, 128, 7, 7366, 9069, 3919, 737, 55, 1, 40689, 49106, 21485, 4304, 380, 11, 238258, 284537, 126273, 26695, 2696, 110, 1, 1483306, 1751554, 785435, 173038, 19272, 976, 16

Offset: 0

Author

Alois P. Heinz, Jan 09 2022

Keywords

• nonn
• tabf

Examples

			T(4,0) = 6: 432|1, 42|31, 42|3|1, 4|31|2, 4|3|21, 4|3|2|1.
T(4,1) = 7: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2.
T(4,2) = 2: 431|2, 41|32.
T(5,2) = 10: 5431|2, 541|32, 531|42, 51|432, 521|4|3, 5|421|3, 5|42|31, 5|42|3|1, 51|4|32, 51|4|3|2.
T(5,3) = 1: 51|42|3.
Triangle T(n,k) begins:
       1;
       0,      1;
       1,      1;
       1,      3,      1;
       6,      7,      2;
      16,     25,     10,     1;
      73,     91,     35,     4;
     298,    390,    163,    25,    1;
    1453,   1797,    755,   128,    7;
    7366,   9069,   3919,   737,   55,   1;
   40689,  49106,  21485,  4304,  380,  11;
  238258, 284537, 126273, 26695, 2696, 110, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Wikipedia, Partition of a set

Crossrefs

Columns k=0-1 give: A350649, A350650.

Row sums give A000110.

T(2n,n) gives A000124(n-1) for n>=1.

T(2n+1,n+1) gives A000012.

Cf. A259691, A350648.

Programs

Maple

b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
      `if`(j=n, x, 1)*b(n-1, max(m, j)), j=1..m+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..ceil(n/2)))(b(n, 0)):
seq(T(n), n=0..14);

Mathematica

b[n_, m_] := b[n, m] = Expand[If[n == 0, 1, Sum[
     If[j == n, x, 1]*b[n-1, Max[m, j]], {j, 1, m+1}]]];
T[n_] := With[{p = b[n, 0]},
Table[Coefficient[p, x, i], {i, 0, Ceiling[n/2]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)

Formula

Sum_{k=1..ceiling(n/2)} k * T(n,k) = A350648(n).