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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
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-François Alcover, Jun 03 2018, from Maple *)
CROSSREFS
Main diagonal gives A293114.
Sequence in context: A255636 A292085 A262163 * A306910 A112185 A192062
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 30 2017
STATUS
approved

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)