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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

LINKS

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

FORMULA

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

EXAMPLE

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;

  ...

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-Fran├žois Alcover, Feb 09 2021, after Alois P. Heinz *)

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.

Sequence in context: A201947 A098816 A214639 * A216973 A061314 A193383

Adjacent sequences:  A319492 A319493 A319494 * A319496 A319497 A319498

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 20 2018

STATUS

approved

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Last modified October 24 02:02 EDT 2021. Contains 348217 sequences. (Running on oeis4.)