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A293109 Number T(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter such that within each prefix of a 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. 14
1, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 5, 10, 4, 1, 0, 7, 24, 17, 5, 1, 0, 11, 62, 58, 26, 6, 1, 0, 15, 140, 193, 107, 37, 7, 1, 0, 22, 329, 603, 439, 178, 50, 8, 1, 0, 30, 725, 1852, 1663, 852, 275, 65, 9, 1, 0, 42, 1631, 5539, 6283, 3767, 1500, 402, 82, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Rows n = 0..40, flattened

FORMULA

T(n,k) = A293108(n,k) - A293108(n,k-1) for k>0, T(n,0) = A293108(n,0).

EXAMPLE

Triangle T(n,k) begins:

  1;

  0,  1;

  0,  2,   1;

  0,  3,   3,   1;

  0,  5,  10,   4,   1;

  0,  7,  24,  17,   5,   1;

  0, 11,  62,  58,  26,   6,  1;

  0, 15, 140, 193, 107,  37,  7, 1;

  0, 22, 329, 603, 439, 178, 50, 8, 1;

MAPLE

h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]

    <j, 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):

g:= proc(n, i, l) option remember;

      `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),

        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))

    end:

A:= proc(n, k) option remember; `if`(n=0, 1, add(add(g(d, k, [])

      *d, 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

h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] < j, 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, n}]][Length[l]];

g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Table[1, n]]], g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Append[l, i]]]]]];

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[g[d, k, {}]*d, {d, Divisors[j]}]*A[n - j, k], {j, 1, n}]/n];

T[n_, 0] := A[n, 0]; T[n_, k_] := A[n, k] - A[n, k - 1];

Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 09 2018, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000007, A000041 (for n>0), A293797, A293798, A293799, A293800, A293801, A293802, A293803, A293804, A293805.

Row sums give A293110.

T(2n,n) gives A293111.

Cf. A182172, A293108, A293113.

Sequence in context: A107238 A258170 A055830 * A233530 A079123 A121548

Adjacent sequences:  A293106 A293107 A293108 * A293110 A293111 A293112

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 30 2017

STATUS

approved

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Last modified August 15 00:08 EDT 2022. Contains 356122 sequences. (Running on oeis4.)