A319495 Number T(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that for k>0 the k-th letter occurs at least once and within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 2, 0, 3, 5, 6, 0, 5, 20, 18, 24, 0, 7, 46, 86, 84, 120, 0, 11, 137, 347, 456, 480, 720, 0, 15, 313, 1216, 2136, 2940, 3240, 5040, 0, 22, 836, 4253, 11128, 15300, 22200, 25200, 40320, 0, 30, 1908, 15410, 44308, 90024, 127680, 191520, 221760, 362880

Offset: 0

Author

Alois P. Heinz, Sep 20 2018

Keywords

• nonn
• tabl

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k <= n. T(n,k) = 0 for k > n.

Examples

			T(3,1) = 3: {aaa}, {aa,a}, {a,a,a}.
T(3,2) = 5: {aab}, {aba}, {baa}, {ab,a}, {ba,a}.
T(3,3) = 6: {abc}, {acb}, {bac}, {bca}, {cab}, {cba}.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,   2;
  0,  3,   5,    6;
  0,  5,  20,   18,    24;
  0,  7,  46,   86,    84,   120;
  0, 11, 137,  347,   456,   480,   720;
  0, 15, 313, 1216,  2136,  2940,  3240,  5040;
  0, 22, 836, 4253, 11128, 15300, 22200, 25200, 40320;
  ...
		

Links

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

Crossrefs

Columns k=0-1 give: A000007, A000041 (for n>0).

Row sums give A292713.

Main diagonal gives A000142.

First lower diagonal gives A038720.

Cf. A292712, A319498.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      g(d, k), d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!,
     Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*
     g[d, k], {d, Divisors[j]}]*A[n - j, k], {j, 1, n}]/n];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 09 2021, after Alois P. Heinz *)

Formula

T(n,k) = A292712(n,k) - A292712(n,k-1) for k > 0, T(n,0) = A000007(n).