A309973 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 3, 0, 2, 6, 10, 0, 2, 21, 42, 47, 0, 3, 42, 177, 264, 246, 0, 4, 90, 619, 1746, 2095, 1602, 0, 5, 176, 1809, 7556, 16085, 16608, 11481, 0, 6, 348, 5211, 32621, 100030, 171480, 154385, 95503, 0, 8, 640, 13961, 120964, 522890, 1262832, 1842659, 1503232, 871030

Offset: 0

Author

Alois P. Heinz, Sep 21 2019

Keywords

• nonn
• tabl

Examples

			T(3,1) = 2: 3aaa, 2aa1a.
T(3,2) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
T(3,3) = 10: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   3;
  0, 2,   6,   10;
  0, 2,  21,   42,    47;
  0, 3,  42,  177,   264,    246;
  0, 4,  90,  619,  1746,   2095,   1602;
  0, 5, 176, 1809,  7556,  16085,  16608,  11481;
  0, 6, 348, 5211, 32621, 100030, 171480, 154385, 95503;
  ...
		

Links

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

Crossrefs

Columns k=0-2 give: A000007, A000009 (for n>0), A327890.

Main diagonal gives A005651.

Row sums give A327679.

T(2n,n) gives A327681.

Cf. A327116, A327680.

Programs

Maple

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

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i j, Min[n - i j, i-1], k] Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2020, after Maple *)

Formula

Sum_{k=1..n} k * T(n,k) = A327680(n).