A327244 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 2, 0, 3, 10, 8, 0, 3, 27, 54, 31, 0, 5, 70, 255, 336, 147, 0, 11, 223, 1222, 2692, 2580, 899, 0, 13, 508, 4467, 15512, 25330, 19566, 5777, 0, 19, 1193, 15540, 78819, 194075, 248976, 160377, 41024, 0, 27, 2822, 52981, 375440, 1303250, 2463534, 2593339, 1430288, 322488

Offset: 0

Author

Alois P. Heinz, Sep 14 2019

Keywords

• nonn
• tabl

Examples

			T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.
T(3,2) = 10: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab.
T(3,3) = 8: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    2;
  0,  3,   10,     8;
  0,  3,   27,    54,    31;
  0,  5,   70,   255,   336,    147;
  0, 11,  223,  1222,  2692,   2580,    899;
  0, 13,  508,  4467, 15512,  25330,  19566,   5777;
  0, 19, 1193, 15540, 78819, 194075, 248976, 160377, 41024;
  ...
		

Links

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

Crossrefs

Columns k=0-2 give: A000007, A032020 (for n>0), A327841.

Main diagonal gives A120774.

Row sums give A309670.

T(2n,n) gives A327596.

Cf. A327245, A327595.

Programs

Maple

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1,i),j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p+j]/j!*c[c[k + i - 1, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2020, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A327595(n).