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

Author

Alois P. Heinz, Sep 30 2017

Keywords

• nonn
• tabl

Examples

			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, ...
		

Links

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

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.

Programs

Maple

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

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.