A270701 Total sum T(n,k) of the sizes of all blocks with maximal element k in all set partitions of {1,2,...,n}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 1, 3, 2, 4, 9, 5, 9, 16, 30, 15, 25, 41, 67, 112, 52, 82, 127, 195, 299, 463, 203, 307, 456, 670, 979, 1429, 2095, 877, 1283, 1845, 2623, 3702, 5204, 7307, 10279, 4140, 5894, 8257, 11437, 15717, 21485, 29278, 39848, 54267, 21147, 29427, 40338, 54692, 73561, 98367, 131007, 174029, 230884, 306298

Offset

1,3

Links

Alois P. Heinz, Rows n = 1..141, flattened

Wikipedia, Partition of a set

Formula

T(n,k) = A270702(n,n-k+1).

Example

Row n=3 is [2, 4, 9] = [0+0+0+1+1, 0+2+1+0+1, 3+1+2+2+1] because the set partitions of {1,2,3} are: 123, 12|3, 13|2, 1|23, 1|2|3.
Triangle T(n,k) begins:
:     1;
:     1,    3;
:     2,    4,    9;
:     5,    9,   16,    30;
:    15,   25,   41,    67,   112;
:    52,   82,  127,   195,   299,   463;
:   203,  307,  456,   670,   979,  1429,  2095;
:   877, 1283, 1845,  2623,  3702,  5204,  7307, 10279;
:  4140, 5894, 8257, 11437, 15717, 21485, 29278, 39848, 54267;

Maple

b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(
     `if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,
      [0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,
     `if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))
    end:
T:= (n, k)-> b(n, 0, max(0, 1+n-k))[2]:
seq(seq(T(n, k), k=1..n), n=1..12);

Mathematica

b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m+1, 0, Function[p, p + If[j == -t || t == 1 && j == m+1, {0, p[[1]]}, 0]][b[ n-1, Max[m, j], If[t == 1 && j == m+1, -j, If[t < 0, t, If[t > 0, t-1, 0] ]]]]], {j, 1, m+1}]];
T[n_, k_] := b[n, 0, Max[0, 1+n-k]][[2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 24 2016, translated from Maple *)

Crossrefs

Columns k=1-10 give: A000110(n-1), A270756, A270757, A270758, A270759, A270760, A270761, A270762, A270763, A270764.

Main and lower diagonals give: A124427, A270765, A270766, A270767, A270768, A270769, A270770, A270771, A270772, A270773.

Row sums give A070071.

Reflected triangle gives A270702.

T(2n-1,n) gives A270703.

Sequence in context: A201838 A099257 A325909 * A083762 A173028 A360415

Adjacent sequences: A270698 A270699 A270700 * A270702 A270703 A270704

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 21 2016

Status

approved