A321878 Number T(n,k) of partitions of n into colored blocks of equal parts, such that all colors from a set of size k are used and the colors are introduced in increasing order; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

1, 0, 1, 0, 2, 0, 3, 1, 0, 5, 2, 0, 7, 5, 0, 11, 9, 1, 0, 15, 17, 2, 0, 22, 28, 5, 0, 30, 47, 10, 0, 42, 74, 21, 1, 0, 56, 116, 37, 2, 0, 77, 175, 67, 5, 0, 101, 263, 112, 10, 0, 135, 385, 187, 20, 0, 176, 560, 302, 40, 1, 0, 231, 800, 479, 72, 2, 0, 297, 1135, 741, 127, 5

Offset: 0

Author

Alois P. Heinz, Aug 27 2019

Keywords

• nonn
• look
• tabf

Comments

T(n,k) is defined for all n>=0 and k>=0. The triangle contains only elements with 0 <= k <= A003056(n). T(n,k) = 0 for k > A003056(n).

For fixed k>=1, T(n,k) ~ exp(sqrt(2*(Pi^2 - 6*polylog(2, 1-k))*n/3)) * sqrt(Pi^2 - 6*polylog(2, 1-k)) / (4*k!*sqrt(3*k)*Pi*n). - Vaclav Kotesovec, Sep 18 2019

Examples

			T(6,1) = 11: 111111a, 2a1111a, 22a11a, 222a, 3a111a, 3a2a1a, 33a, 4a11a, 4a2a, 5a1a, 6a.
T(6,2) = 9: 2a1111b, 22a11b, 3a111b, 3a2a1b, 3a2b1a, 3a2b1b, 4a11b, 4a2b, 5a1b.
T(6,3) = 1: 3a2b1c.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2;
  0,  3,   1;
  0,  5,   2;
  0,  7,   5;
  0, 11,   9,  1;
  0, 15,  17,  2;
  0, 22,  28,  5;
  0, 30,  47, 10;
  0, 42,  74, 21, 1;
  0, 56, 116, 37, 2;
  0, 77, 175, 67, 5;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..1000, flattened

Crossrefs

Columns k=0-10 give: A000007, A000041 (for n>0), A327285, A327286, A327287, A327288, A327289, A327290, A327291, A327292, A327293.

Row sums give A305106.

Cf. A000096, A000217, A000712, A003056, A321884, A322304.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k))(n-i*j), j=1..n/i)*k+b(n, i-1, k)))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
Table[Table[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Formula

T(n,k) = 1/k! * Sum_{i=0..k} (-1)^i*binomial(k,i) A321884(n,k-i).

T(n*(n+1)/2,n) = T(A000217(n),n) = 1.

T(n*(n+3)/2,n) = T(A000096(n),n) = A000712(n).

Sum_{k=1..A003056(n)} k * T(n,k) = A322304(n).