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

0,7

Links

Alois P. Heinz, Rows n = 0..200, flattened

Wikipedia, Partition of a set

Formula

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

Example

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;
  ...

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 *)

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.

Sequence in context: A359574 A210749 A330587 * A199662 A280293 A226483

Adjacent sequences: A350644 A350645 A350646 * A350648 A350649 A350650

Keyword

nonn,tabf

Author

Alois P. Heinz, Jan 09 2022

Status

approved