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A293113
Number T(n,k) of sets 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, 1, 1, 0, 2, 3, 1, 0, 2, 8, 4, 1, 0, 3, 20, 16, 5, 1, 0, 4, 47, 53, 25, 6, 1, 0, 5, 106, 173, 102, 36, 7, 1, 0, 6, 237, 532, 410, 172, 49, 8, 1, 0, 8, 522, 1615, 1545, 813, 268, 64, 9, 1, 0, 10, 1146, 4785, 5784, 3576, 1448, 394, 81, 10, 1
OFFSET
0,8
LINKS
FORMULA
T(n,k) = A293112(n,k) - A293112(n,k-1) for k>0, T(n,0) = A293112(n,0).
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 3, 1;
0, 2, 8, 4, 1;
0, 3, 20, 16, 5, 1;
0, 4, 47, 53, 25, 6, 1;
0, 5, 106, 173, 102, 36, 7, 1;
0, 6, 237, 532, 410, 172, 49, 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:
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:
T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
seq(seq(T(n, k), k=0..n), n=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}]]];
T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A000007, A000009 (for n>0), A293883, A293884, A293885, A293886, A293887, A293888, A293889, A293890, A293891.
Row sums give A293114.
T(2n,n) gives A293115.
Sequence in context: A372257 A253580 A020921 * A366528 A154720 A355487
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 30 2017
STATUS
approved