A350684 Number T(n,k) of partitions of [n] such that the sum of elements i contained in block i equals k when blocks are ordered with decreasing largest elements; triangle T(n,k), n>=0, 0<=k<=max(0,A008805(n-1)), read by rows.

1, 0, 1, 1, 1, 1, 1, 2, 1, 6, 3, 4, 2, 16, 7, 8, 14, 3, 3, 1, 73, 25, 26, 51, 12, 12, 4, 298, 91, 92, 164, 116, 56, 30, 21, 4, 4, 1, 1453, 390, 391, 601, 676, 256, 163, 147, 28, 28, 7, 7366, 1797, 1798, 2484, 3228, 1927, 897, 876, 307, 307, 87, 31, 31, 5, 5, 1

Offset

0,8

Links

Alois P. Heinz, Rows n = 0..100, flattened

Wikipedia, Partition of a set

Formula

Sum_{k=1..max(0,A008805(n-1))} k * T(n,k) = A350683(n).

T(2n,A000217(n)) = A152947(n+1).

T(2n-1,A000217(n)) = 1 for n>=1.

T(n,2) - T(n,1) = 1 for n>=3.

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) = 3: 432(1), 42(1)|3, 4(1)|3|2.
T(4,2) = 4: 43|(2)1, 43|(2)|1, 4|3(2)1, 4|3(2)|1,
T(4,3) = 2: 43(1)|(2), 4(1)|3(2).
Triangle T(n,k) begins:
     1;
     0,   1;
     1,   1;
     1,   1,   2,   1;
     6,   3,   4,   2;
    16,   7,   8,  14,   3,   3,   1;
    73,  25,  26,  51,  12,  12,   4;
   298,  91,  92, 164, 116,  56,  30,  21,  4,  4, 1;
  1453, 390, 391, 601, 676, 256, 163, 147, 28, 28, 7;
  ...

Maple

b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
     `if`(n=j, x^j, 1)*b(n-1, max(m, j)), j=1..m+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);

Mathematica

b[n_, m_] := b[n, m] = Expand[If[n == 0, 1, Sum[
     If[n == j, x^j, 1]*b[n - 1, Max[m, j]], {j, 1, m + 1}]]];
T[n_] := CoefficientList[b[n, 0], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

Crossrefs

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

Row sums give A000110.

Cf. A000217, A008805, A152947, A350647, A350683.

Sequence in context: A131449 A289869 A334595 * A124443 A077172 A160047

Adjacent sequences: A350681 A350682 A350683 * A350685 A350686 A350687

Keyword

nonn,tabf

Author

Alois P. Heinz, Jan 11 2022

Status

approved