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 *)
A293737
Number of multisets 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, 3, 7, 20, 54, 164, 500, 1629, 5462, 19164, 69457, 261154, 1012164, 4045640, 16611121, 70001515, 301922104, 1331128134, 5986321599, 27426419974, 127801386949, 605016657100, 2906093083727, 14149469612919, 69762426194708, 348016146152252, 1755188873640756
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:
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);
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
A293889
Number of sets 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, 81, 554, 3723, 22795, 138112, 804518, 4666521, 26601027, 151607448, 858078020, 4867646261, 27569926921, 156731332711, 892188573587, 5101471557918, 29254697416411, 168575093334130, 975019430127356, 5667406033511452, 33079212453456235, 194029375272192986
Offset: 8
Showing 1-4 of 4 results.
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