A364310 Number T(n,k) of partitions of n into k parts where each block of part i with multiplicity j is marked with a word of length i*j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 15, 15, 10, 1, 0, 1, 22, 76, 35, 15, 1, 0, 1, 63, 168, 252, 70, 21, 1, 0, 1, 93, 574, 785, 658, 126, 28, 1, 0, 1, 255, 2188, 3066, 2739, 1470, 210, 36, 1, 0, 1, 386, 5490, 18235, 12181, 7857, 2940, 330, 45, 1

Offset

0,9

Links

Alois P. Heinz, Rows n = 0..150, flattened

Example

T(4,1) = 1: 4abcd.
T(4,2) = 5: 3abc1d, 3abd1c, 3acd1b, 3bcd1a, 22abcd.
T(4,3) = 6: 2ab11cd, 2ac11bd, 2ad11bc, 2bc11ad, 2bd11ac, 2cd11ab.
T(4,4) = 1: 1111abcd.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   3,    1;
  0, 1,   5,    6,    1;
  0, 1,  15,   15,   10,    1;
  0, 1,  22,   76,   35,   15,    1;
  0, 1,  63,  168,  252,   70,   21,   1;
  0, 1,  93,  574,  785,  658,  126,  28,  1;
  0, 1, 255, 2188, 3066, 2739, 1470, 210, 36, 1;
  ...

Maple

b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*x^j*binomial(n, i*j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
   Sum[b[n - i*j, i - 1]*x^j*Binomial[n, i*j], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 18 2023, after Alois P. Heinz *)

Crossrefs

Columns k=0-1 give: A000007, A057427.

Row sums give A178682.

T(n,n) gives A000012.

T(n+1,n) gives A000217.

T(n+2,n) gives A000332(n+3).

Cf. A364285.

Sequence in context: A327618 A121314 A119271 * A323222 A125104 A098157

Adjacent sequences: A364307 A364308 A364309 * A364311 A364312 A364313

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 18 2023

Status

approved