A182930 Triangle read by rows: Number of set partitions of {1,2,..,n} such that |k| is a block and no block |m| with m < k exists, (1 <= n, 1 <= k <= n).

1, 1, 0, 2, 1, 1, 5, 3, 2, 1, 15, 10, 7, 5, 4, 52, 37, 27, 20, 15, 11, 203, 151, 114, 87, 67, 52, 41, 877, 674, 523, 409, 322, 255, 203, 162, 4140, 3263, 2589, 2066, 1657, 1335, 1080, 877, 715, 21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140, 3425

Offset: 1

Author

Peter Luschny, Apr 08 2011

Keywords

• nonn
• tabl

Comments

Mirror image of A106436. - Alois P. Heinz, Jan 29 2019

Examples

			T(4,2) = card({2|134, 2|3|14, 2|4|13}) = 3.
[1]     1,
[2]     1,    0,
[3]     2,    1,    1,
[4]     5,    3,    2,    1,
[5]    15,   10,    7,    5,    4,
[6]    52,   37,   27,   20,   15,   11,
     [-1-] [-2-] [-3-] [-4-] [-5-] [-6-]
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Peter Luschny, Set partitions

Crossrefs

Cf. A000110, A000296, A106436.

T(2n+1,n+1) gives A020556.

Programs

Maple

T := proc(n, k) option remember; if n = 1 then 1 elif n = k then T(n-1,1) - T(n-1,n-1) else T(n-1,k) + T(n, k+1) fi end:
A182930 := (n,k) -> T(n,k); seq(print(seq(A182930(n,k),k=1..n)),n=1..6);

Mathematica

T[n_, k_] := T[n, k] = Which[n == 1, 1, n == k, T[n-1, 1] - T[n-1, n-1], True, T[n-1, k] + T[n, k+1]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* Jean-François Alcover, Jun 22 2019 *)

Formula

Recursion: The value of T(n,k) is, if n < 0 or k < 0 or k > n undefined, else if n = 1 then 1 else if k = n then T(n-1,1) - T(n-1,n-1); in all other cases T(n,k) = T(n,k+1) + T(n-1,k).