A319375 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 10, 4, 1, 35, 17, 7, 1, 136, 76, 36, 11, 1, 577, 357, 186, 81, 16, 1, 2682, 1737, 1023, 512, 162, 22, 1, 13435, 8997, 5867, 3151, 1345, 295, 29, 1, 72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1, 414761, 289253, 211888, 133853, 72973, 29503, 6676, 796, 46, 1

Offset

1,2

Links

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

Wikipedia, Partition of a set

Example

The 5 set partitions of {1,2,3} are:
  1   |2  |3
  12  |3
  13  |2
  23  |1
  123
so there are 10 elements in the first (largest) blocks, 4 in the second blocks and only 1 in the third blocks.
Triangle T(n,k) begins:
      1;
      3,     1;
     10,     4,     1;
     35,    17,     7,     1;
    136,    76,    36,    11,     1;
    577,   357,   186,    81,    16,    1;
   2682,  1737,  1023,   512,   162,   22,   1;
  13435,  8997,  5867,  3151,  1345,  295,  29,  1;
  72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1;

Maple

b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*
      x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
      b(n-j, sort([l[], j])), j=1..n))
    end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
       combinat[multinomial](n, i$j, n-i*j)/j!*
      b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
    end:
T:= (n, k)-> b(n$2, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 02 2020

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[-i]] x^i, {i, 1, Length[l]}], Sum[ Binomial[n-1, j-1] b[n-j, Sort[Append[l, j]]], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, {}]];
Array[T, 12] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *)

Crossrefs

Columns k=1-10 give: A097148, A333059, A333060, A333061, A333062, A333063, A333064, A333065, A333066, A333067.

Row sums give A070071.

Cf. A319298, A322384.

Sequence in context: A280787 A126954 A176992 * A107870 A078817 A316193

Adjacent sequences: A319372 A319373 A319374 * A319376 A319377 A319378

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 07 2018

Status

approved