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 *)
A293745
Number of sets of nonempty words with a total of n letters over senary 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, 429, 1406, 4771, 16749, 60453, 224948, 857010, 3350574, 13366375, 54494538, 226020624, 954737292, 4092229831, 17813005015, 78509835288, 350592604663, 1582430253294, 7223028969003, 33275812688050, 154790795962448, 725871751770492
Offset: 0
-
g:= proc(n) option remember;
`if`(n<4, [1, 1, 2, 4][n+1], ((20*n^2+184*n+336)*g(n-1)
+4*(n-1)*(10*n^2+58*n+33)*g(n-2) -144*(n-1)*(n-2)*g(n-3)
-144*(n-1)*(n-2)*(n-3)*g(n-4))/ ((n+5)*(n+8)*(n+9)))
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, 6];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 06 2018, using code from A293112 *)
A293801
Number of multisets of nonempty words with a total of n letters over senary alphabet containing the sixth 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, 7, 50, 275, 1500, 7517, 37481, 180552, 868974, 4120598, 19574576, 92438492, 437977946, 2072100450, 9843499790, 46801485700, 223513049582, 1069698339831, 5142776951972, 24793644707917, 120072824665722, 583316549714731, 2846266161931683, 13933753316683787
Offset: 6
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
Showing 1-4 of 4 results.