A327884 Number T(n,k) of set partitions of [n] such that at least one of the block sizes is k or k=0; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 2, 1, 1, 5, 4, 3, 1, 15, 11, 9, 4, 1, 52, 41, 35, 20, 5, 1, 203, 162, 150, 90, 30, 6, 1, 877, 715, 672, 455, 175, 42, 7, 1, 4140, 3425, 3269, 2352, 1015, 280, 56, 8, 1, 21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1, 115975, 98253, 97155, 76540, 39480, 12978, 3150, 600, 90, 10, 1

Offset

0,4

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Iverson bracket

Wikipedia, Partition of a set

Formula

E.g.f. of column k: exp(exp(x)-1) - [k>0] * exp(exp(x)-1-x^k/k!).

T(n,0) - T(n,1) = A000296(n).

Example

T(4,1) = 11: 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
T(4,3) = 4: 123|4, 124|3, 134|2, 1|234.
T(4,4) = 1: 1234.
T(5,1) = 41: 1234|5, 1235|4, 123|4|5, 1245|3, 124|3|5, 12|34|5, 125|3|4, 12|35|4, 12|3|45, 12|3|4|5, 1345|2, 134|2|5, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 1|23|4|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
Triangle T(n,k) begins:
      1;
      1,     1;
      2,     1,     1;
      5,     4,     3,     1;
     15,    11,     9,     4,    1;
     52,    41,    35,    20,    5,    1;
    203,   162,   150,    90,   30,    6,   1;
    877,   715,   672,   455,  175,   42,   7,  1;
   4140,  3425,  3269,  2352, 1015,  280,  56,  8, 1;
  21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1;
  ...

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
    end:
T:= (n, k)-> b(n, 0)-`if`(k=0, 0, b(n, k)):
seq(seq(T(n, k), k=0..n), n=0..11);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[If[j == k, 0, b[n - j, k] Binomial[ n - 1, j - 1]], {j, 1, n}]];
T[n_, k_] := b[n, 0] - If[k == 0, 0, b[n, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-3 give: A000110, A000296(n+1), A327885, A328153.

T(2n,n) gives A276961.

Cf. A080510, A327869.

Sequence in context: A284732 A371766 A327483 * A050145 A222573 A222679

Adjacent sequences: A327881 A327882 A327883 * A327885 A327886 A327887

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 28 2019

Status

approved