A327673 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 arbitrary order); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 3, 0, 3, 18, 19, 0, 3, 60, 171, 121, 0, 5, 210, 1173, 1996, 1041, 0, 11, 798, 7512, 22784, 27225, 11191, 0, 13, 2462, 39708, 196904, 411115, 382086, 130663, 0, 19, 7891, 204987, 1546042, 4991815, 7843848, 5932843, 1731969

Offset

0,6

Links

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

Formula

Sum_{k=1..n} k * T(n,k) = A327676(n).

Example

T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.
T(3,2) = 18: 3aab, 3aba, 3baa, 3abb, 3bab, 3bba, 2aa1b, 2ab1a, 2ba1a, 2ab1b, 2ba1b, 2bb1a, 1a2ab, 1a2ba, 1a2bb, 1b2aa, 1b2ab, 1b2ba.
T(3,3) = 19: 3abc, 3acb, 3bac, 3bca, 3cab, 3cba, 2ab1c, 2ac1b, 2ba1c, 2bc1a, 2ca1b, 2cb1a, 1a2bc, 1a2cb, 1b2ac, 1b2ca, 1c2ab, 1c2ba, 1a1b1c.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    3;
  0,  3,   18,    19;
  0,  3,   60,   171,    121;
  0,  5,  210,  1173,   1996,   1041;
  0, 11,  798,  7512,  22784,  27225,  11191;
  0, 13, 2462, 39708, 196904, 411115, 382086, 130663;
  ...

Maple

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

Mathematica

b[n_, i_, k_, p_] := b[n, i, k, p] = If[n==0, p!, If[i<1, 0, Sum[Binomial[ k^i, j] b[n - i j, Min[n - i j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i, 0] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 30 2020, after Maple *)

Crossrefs

Columns k=0-2 give: A000007, A032020 (for n>0), A327768.

Main diagonal gives A327674.

Row sums give A327675.

T(2n,n) gives A327678.

Cf. A327244, A327676.

Sequence in context: A275080 A128252 A230675 * A279657 A272722 A229694

Adjacent sequences: A327670 A327671 A327672 * A327674 A327675 A327676

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 21 2019

Status

approved