A293108 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 3, 0, 1, 1, 3, 6, 5, 0, 1, 1, 3, 7, 15, 7, 0, 1, 1, 3, 7, 19, 31, 11, 0, 1, 1, 3, 7, 20, 48, 73, 15, 0, 1, 1, 3, 7, 20, 53, 131, 155, 22, 0, 1, 1, 3, 7, 20, 54, 157, 348, 351, 30, 0, 1, 1, 3, 7, 20, 54, 163, 455, 954, 755, 42, 0

Offset

0,9

Links

Alois P. Heinz, Antidiagonals n = 0..80, flattened

Formula

G.f. of column k: Product_{j>=1} 1/(1-x^j)^A182172(j,k).

A(n,k) = Sum_{j=0..k} A293109(n,j).

A(n,n) = A(n,k) for all k >= n.

Example

Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,   1,   1,   1,   1,   1,   1, ...
  0,  2,   3,   3,   3,   3,   3,   3, ...
  0,  3,   6,   7,   7,   7,   7,   7, ...
  0,  5,  15,  19,  20,  20,  20,  20, ...
  0,  7,  31,  48,  53,  54,  54,  54, ...
  0, 11,  73, 131, 157, 163, 164, 164, ...
  0, 15, 155, 348, 455, 492, 499, 500, ...

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:
seq(seq(A(n, d-n), n=0..d), d=0..14);

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];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

Crossrefs

Columns k=0-10 give: A000007, A000041, A293732, A293733, A293734, A293735, A293736, A293737, A293738, A293739, A293740.

Main diagonal gives A293110.

Cf. A182172, A293109, A293112.

Sequence in context: A219782 A035788 A097559 * A172237 A246181 A123226

Adjacent sequences: A293105 A293106 A293107 * A293109 A293110 A293111

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 30 2017

Status

approved