A262071 Number T(n,k) of ordered partitions of an n-set with nondecreasing block sizes and maximal block size equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 6, 3, 1, 0, 24, 18, 4, 1, 0, 120, 90, 30, 5, 1, 0, 720, 630, 200, 45, 6, 1, 0, 5040, 4410, 1610, 350, 63, 7, 1, 0, 40320, 37800, 13440, 3290, 560, 84, 8, 1, 0, 362880, 340200, 130200, 30870, 5922, 840, 108, 9, 1, 0, 3628800, 3515400, 1327200, 334950, 61992, 9870, 1200, 135, 10, 1

Offset

0,5

Links

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

Formula

E.g.f. of column k: x^k * Product_{i=1..k} (i-1)!/(i!-x^i).

Example

T(3,1) = 6: 1|2|3, 1|3|2, 2|1|3, 2|3|1, 3|1|2, 3|2|1.
T(3,2) = 3: 1|23, 2|13, 3|12.
T(3,3) = 1: 123.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2,     1;
  0,     6,     3,     1;
  0,    24,    18,     4,    1;
  0,   120,    90,    30,    5,   1;
  0,   720,   630,   200,   45,   6,  1;
  0,  5040,  4410,  1610,  350,  63,  7, 1;
  0, 40320, 37800, 13440, 3290, 560, 84, 8, 1;

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
    end:
T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, Binomial[n, i]*b[n - i, i]]]]; T[n_, k_] :=  b[n, k] - If[k == 0, 0, b[n, k - 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 12 2016, Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A000142 (for n>0), A272492, A272493, A272494, A272495, A272496, A272497, A272498, A272499, A272500.

Main diagonal gives A000012.

Row sums give A005651.

T(2n,n) gives A266518.

Cf. A262072.

Sequence in context: A180663 A331327 A301924 * A011312 A275328 A147720

Adjacent sequences: A262068 A262069 A262070 * A262072 A262073 A262074

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 10 2015

Status

approved