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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. 14
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..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-Fran├žois 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: 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 October 3 07:51 EDT 2022. Contains 357231 sequences. (Running on oeis4.)