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A293112 Number A(n,k) of sets 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. 13
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 5, 2, 0, 1, 1, 2, 6, 10, 3, 0, 1, 1, 2, 6, 14, 23, 4, 0, 1, 1, 2, 6, 15, 39, 51, 5, 0, 1, 1, 2, 6, 15, 44, 104, 111, 6, 0, 1, 1, 2, 6, 15, 45, 129, 284, 243, 8, 0, 1, 1, 2, 6, 15, 45, 135, 386, 775, 530, 10, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

LINKS

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

FORMULA

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

A(n,k) = Sum_{j=0..k} A293113(n,j).

EXAMPLE

Square array A(n,k) begins:

  1, 1,   1,   1,   1,   1,   1,   1, ...

  0, 1,   1,   1,   1,   1,   1,   1, ...

  0, 1,   2,   2,   2,   2,   2,   2, ...

  0, 2,   5,   6,   6,   6,   6,   6, ...

  0, 2,  10,  14,  15,  15,  15,  15, ...

  0, 3,  23,  39,  44,  45,  45,  45, ...

  0, 4,  51, 104, 129, 135, 136, 136, ...

  0, 5, 111, 284, 386, 422, 429, 430, ...

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:

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(b(n-i*j, i-1, k)*binomial(g(i, k, []), j), j=0..n/i)))

    end:

A:= (n, k)-> b(n$2, k):

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, 1, 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]]]]]];

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*Binomial[g[i, k, {}], j], {j, 0, n/i}]]];

A[n_, k_] := b[n, n, k];

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

CROSSREFS

Columns k=0-10 give: A000007, A000009, A293741, A293742, A293743, A293744, A293745, A293746, A293747, A293748, A293749.

Main diagonal gives A293114.

Cf. A182172, A293108, A293113.

Sequence in context: A255636 A292085 A262163 * A306910 A112185 A192062

Adjacent sequences:  A293109 A293110 A293111 * A293113 A293114 A293115

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 30 2017

STATUS

approved

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Last modified May 9 20:41 EDT 2021. Contains 343746 sequences. (Running on oeis4.)