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

0,5

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

Links

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

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).

Example

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;
  ...

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 *)

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.

Sequence in context: A144955 A225624 A168020 * A225084 A238345 A299070

Adjacent sequences: A321875 A321876 A321877 * A321879 A321880 A321881

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Aug 27 2019

Status

approved