A319298 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 7, 7, 1, 21, 25, 13, 1, 66, 101, 71, 21, 1, 258, 366, 396, 166, 31, 1, 1079, 1555, 1877, 1247, 337, 43, 1, 4987, 7099, 9199, 7855, 3305, 617, 57, 1, 25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1, 136723, 184033, 253108, 284968, 203278, 79756, 16126, 1666, 91, 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
  1   |23
  2   |13
  3   |12
  123
so there are 7 elements in the first (smallest) blocks, 7 in the second blocks and only 1 in the third blocks.
Triangle T(n,k) begins:
      1;
      3,     1;
      7,     7,     1;
     21,    25,    13,     1;
     66,   101,    71,    21,     1;
    258,   366,   396,   166,    31,    1;
   1079,  1555,  1877,  1247,   337,   43,    1;
   4987,  7099,  9199,  7855,  3305,  617,   57,  1;
  25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 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>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
      max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
    end:
T:= (n, k)-> b(n, 1, 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, {}]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *)

Crossrefs

Column k=1-10 gives A097147, A332942, A332943, A332944, A332945, A332946, A332947, A332948, A332949, A332950.

Row sums give A070071.

Cf. A319375, A322383.

Sequence in context: A188463 A369892 A359576 * A101748 A058606 A135284

Adjacent sequences: A319295 A319296 A319297 * A319299 A319300 A319301

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 07 2018

Status

approved