A261836 Number T(n,k) of compositions of n into distinct parts where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the composition; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 3, 10, 7, 0, 3, 15, 21, 9, 0, 5, 40, 96, 92, 31, 0, 11, 183, 832, 1562, 1305, 403, 0, 13, 266, 1539, 3908, 4955, 3090, 757, 0, 19, 549, 4281, 14791, 26765, 26523, 13671, 2873, 0, 27, 1056, 10902, 50208, 124450, 178456, 148638, 66904, 12607

Offset

0,8

Comments

Also number of matrices with k rows of nonnegative integer entries and without zero rows or columns such that the sum of all entries is equal to n and the column sums are distinct.

Links

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

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261835(n,k-i).

Example

T(3,2) = 10: (matrices and corresponding marked compositions are given)
  [2]   [1]   [2 0]  [0 2]  [1 0]  [0 1]  [1 1]  [1 1]  [1 0]  [0 1]
  [1]   [2]   [0 1]  [1 0]  [0 2]  [2 0]  [1 0]  [0 1]  [1 1]  [1 1]
  3aab, 3abb, 2aa1b, 1b2aa, 1a2bb, 2bb1a, 2ab1a, 1a2ab, 2ab1b, 1b2ab.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,   1;
  0,  3,  10,    7;
  0,  3,  15,   21,     9;
  0,  5,  40,   96,    92,    31;
  0, 11, 183,  832,  1562,  1305,   403;
  0, 13, 266, 1539,  3908,  4955,  3090,   757;
  0, 19, 549, 4281, 14791, 26765, 26523, 13671, 2873;

Maple

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

Mathematica

b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2<n, 0, If[n==0, p!, b[n, i -1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; T[n_, k_] := Sum[b[n, n, 0, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A032020 (for n>0), A261853, A261854, A261855, A261856, A261857, A261858, A261859, A261860, A261861.

Main diagonal gives A032011.

Row sums give A261838.

T(2n,n) gives A261828.

Cf. A261781, A261835.

Sequence in context: A289832 A196163 A195922 * A301937 A373866 A185139

Adjacent sequences: A261833 A261834 A261835 * A261837 A261838 A261839

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 02 2015

Status

approved