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.
Original entry on oeis.org
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
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, ...
Columns k=0-10 give:
A000007,
A000041,
A293732,
A293733,
A293734,
A293735,
A293736,
A293737,
A293738,
A293739,
A293740.
-
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);
-
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 *)
A293746
Number of sets of nonempty words with a total of n letters over septenary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
Original entry on oeis.org
1, 1, 2, 6, 15, 45, 136, 430, 1414, 4835, 17143, 62843, 238245, 930418, 3741710, 15445815, 65384356, 283113205, 1252393193, 5648731817, 25945636702, 121172059749, 574764521186, 2765620022767, 13486540312370, 66587056756662, 332594340605540, 1679325348600290
Offset: 0
-
g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],
((4*n^3+78*n^2+424*n+495)*g(n-1) +(n-1)*(34*n^2+280*n+
305)*g(n-2) -2*(n-1)*(n-2)*(38*n+145)*g(n-3) -105*(n-1)
*(n-2)*(n-3)*g(n-4))/((n+6)*(n+10)*(n+12)))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
-
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_] := b[n, n, 7];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 06 2018, using code from A293112 *)
A293802
Number of multisets of nonempty words with a total of n letters over septenary alphabet containing the seventh letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
Original entry on oeis.org
1, 8, 65, 402, 2457, 13715, 75997, 405672, 2160487, 11323773, 59410323, 309820614, 1620398321, 8465984310, 44406170067, 233282360743, 1230967142377, 6514406116198, 34634583683433, 184803113451639, 990647642903411, 5331286355111395, 28821503947539335
Offset: 7
A293803
Number of multisets of nonempty words with a total of n letters over octonary alphabet containing the eighth letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
Original entry on oeis.org
1, 9, 82, 563, 3807, 23376, 142085, 829038, 4816390, 27477989, 156716051, 887263747, 5034299169, 28513927121, 162089445027, 922536360293, 5273968531519, 30236181831043, 174184141284927, 1007162425272715, 5852489042303046, 34148843838375080, 200241641820235902
Offset: 8
Showing 1-4 of 4 results.