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

Author

Alois P. Heinz, Dec 07 2018

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• Wikipedia, Partition of a set

Crossrefs

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

Row sums give A070071.

Cf. A319298, A322384.

Programs

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 *)