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.
Original entry on oeis.org
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
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, 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, ...
Columns k=0-10 give:
A000007,
A000009,
A293741,
A293742,
A293743,
A293744,
A293745,
A293746,
A293747,
A293748,
A293749.
-
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:
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);
-
h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] 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 *)
A293736
Number of multisets 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, 3, 7, 20, 54, 164, 499, 1621, 5397, 18762, 67000, 247439, 936167, 3639968, 14450634, 58677742, 242511781, 1021307520, 4365923278, 18960435664, 83395216882, 371734296357, 1675125941350, 7635063496721, 35127842511275, 163213032700613, 764541230737345
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:
a:= proc(n) option remember; `if`(n=0, 1, add(add(g(d)
*d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..35);
A293887
Number of sets 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, 49, 268, 1448, 7228, 35878, 172470, 828380, 3924415, 18627313, 87940196, 416581962, 1971018227, 9364427840, 44535275966, 212749669917, 1018545160923, 4898593478484, 23625691956720, 114460931725922, 556278543737430, 2715406810871312, 13298453769286699
Offset: 6
A293888
Number of sets 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, 64, 394, 2390, 13297, 73408, 391136, 2079440, 10889818, 57092581, 297655901, 1556501986, 8132631687, 42662224461, 224171916523, 1183189769473, 6263511343367, 33311244068612, 177803544643092, 953453596829798, 5132942333604244, 27758886429083153
Offset: 7
Showing 1-4 of 4 results.
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