A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

1, 1, 1, 1, 1, 4, 1, 13, 1, 1, 41, 9, 1, 1, 131, 59, 11, 1, 1, 428, 344, 88, 15, 1, 1, 1429, 1906, 634, 146, 23, 1, 1, 4861, 10345, 4389, 1231, 280, 39, 1, 1, 16795, 55901, 30006, 9835, 2763, 602, 71, 1, 1, 58785, 303661, 205420, 77178, 25014, 6967, 1408, 135, 1

Offset: 0

Author

Alois P. Heinz, May 28 2017

Keywords

• nonn
• tabf

Examples

			T(4,0) = 1: 1234.
T(4,1) = 13: 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 1: 13|2|4.
T(5,2) = 9: 124|3|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|35, 14|2|3|5, 1|24|3|5.
T(6,3) = 11: 1245|3|6, 1346|2|5, 134|26|5, 134|2|56, 134|2|5|6, 145|23|6, 145|2|36, 145|2|3|6, 14|25|3|6, 15|24|3|6, 1|245|3|6.
T(6,4) = 1: 1345|2|6.
T(7,4) = 15: 12456|3|7, 13457|2|6, 1345|27|6, 1345|2|67, 1345|2|6|7, 1456|23|7, 1456|2|37, 1456|2|3|7, 145|26|3|7, 146|25|3|7, 14|256|3|7, 156|24|3|7, 15|246|3|7, 16|245|3|7, 1|2456|3|7.
Triangle T(n,k) begins:
  1;
  1;
  1,    1;
  1,    4;
  1,   13,     1;
  1,   41,     9,    1;
  1,  131,    59,   11,    1;
  1,  428,   344,   88,   15,   1;
  1, 1429,  1906,  634,  146,  23,  1;
  1, 4861, 10345, 4389, 1231, 280, 39, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..20, flattened
• Wikipedia, Partition of a set

Crossrefs

Columns k=0-1 give: A000012, A001453.

Row sums give A000110.

Cf. A168415, A287213, A287215, A287416, A287641.

Programs

Maple

b:= proc(n, l) option remember; `if`(n=0 or l=[], 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
T:= (n, k)-> `if`(k=0, 1, b(n, [0$k])-b(n, [0$k-1])):
seq(seq(T(n, k), k=0..max(n/2, n-2)), n=0..12);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0 || l == {}, 1, Sum[b[n-1, Append[Table[ Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
T[n_, k_] := If[k == 0, 1, b[n, Table[0, k]] - b[n, Table[0, k - 1]]];
Table[T[n, k], {n, 0, 12}, { k, 0, Max[n/2, n - 2]}] // Flatten (* Jean-François Alcover, May 22 2018, translated from Maple *)

Formula

T(n,k) = A287641(n,k) - A287641(n,k-1) for k>0, T(n,0) = 1.

T(n+4,n+1) = A168415(n) for n>0.