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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Rows n = 1..50, flattened

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);

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-Fran├žois Alcover, Dec 28 2018, after Alois P. Heinz *)

CROSSREFS

Column k=1 gives A097147.

Row sums give A070071.

Cf. A319375, A322383.

Sequence in context: A136035 A132307 A188463 * A101748 A058606 A135284

Adjacent sequences:  A319295 A319296 A319297 * A319299 A319300 A319301

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 07 2018

STATUS

approved

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Last modified February 25 07:33 EST 2020. Contains 332221 sequences. (Running on oeis4.)