A293113
Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 2, 8, 4, 1, 0, 3, 20, 16, 5, 1, 0, 4, 47, 53, 25, 6, 1, 0, 5, 106, 173, 102, 36, 7, 1, 0, 6, 237, 532, 410, 172, 49, 8, 1, 0, 8, 522, 1615, 1545, 813, 268, 64, 9, 1, 0, 10, 1146, 4785, 5784, 3576, 1448, 394, 81, 10, 1
Offset: 0
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 2, 3, 1;
0, 2, 8, 4, 1;
0, 3, 20, 16, 5, 1;
0, 4, 47, 53, 25, 6, 1;
0, 5, 106, 173, 102, 36, 7, 1;
0, 6, 237, 532, 410, 172, 49, 8, 1;
...
Columns k=0-10 give:
A000007,
A000009 (for n>0),
A293883,
A293884,
A293885,
A293886,
A293887,
A293888,
A293889,
A293890,
A293891.
-
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:
T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
seq(seq(T(n, k), k=0..n), n=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}]]];
T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)
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 *)
A293110
Number of multisets of nonempty words with a total of n letters over n-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.
Original entry on oeis.org
1, 1, 3, 7, 20, 54, 164, 500, 1630, 5472, 19257, 70133, 265858, 1042346, 4235031, 17760943, 76913277, 342919431, 1573637985, 7415371293, 35860511131, 177641956111, 900782461170, 4668600610346, 24714284921937, 133467868645017, 734844788634269, 4120752558254581
Offset: 0
a(0) = 1: {}.
a(1) = 1: {a}
a(2) = 3: {a,a}, {aa}, {ab}.
a(3) = 7: {a,a,a}, {a,aa}, {a,ab}, {aaa}, {aab}, {aba}, {abc}.
-
g:= proc(n) option remember;
`if`(n<2, 1, g(n-1)+(n-1)*g(n-2))
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..40);
-
g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[g[d]*d, {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 07 2018, from Maple *)
A293111
Number of multisets of nonempty words with a total of 2n letters over n-ary alphabet containing the n-th 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, 2, 10, 58, 439, 3767, 37481, 405672, 4816390, 60748286, 817467196, 11533898929, 170979909037, 2635984019533, 42297026556483, 701880878089205, 12045976560853148, 212911592290588547, 3874514946593004395, 72378921228197309169, 1387364840045160600103
Offset: 0
A293797
Number of multisets of nonempty words with a total of n letters over binary alphabet containing the second 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, 3, 10, 24, 62, 140, 329, 725, 1631, 3548, 7820, 16887, 36767, 79046, 170799, 365883, 786361, 1679486, 3594580, 7656839, 16331933, 34705064, 73810524, 156501393, 332001987, 702530049, 1486995062, 3140712201, 6634309660, 13988510961, 29494808402, 62091823865
Offset: 2
A293798
Number of multisets of nonempty words with a total of n letters over ternary alphabet containing the third 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, 4, 17, 58, 193, 603, 1852, 5539, 16391, 47919, 139258, 402365, 1159175, 3330974, 9561080, 27421882, 78642072, 225566206, 647308614, 1858741431, 5341591906, 15363474607, 44228744168, 127444421304, 367573522408, 1061131077602, 3066124030310, 8867379519154
Offset: 3
A293799
Number of multisets of nonempty words with a total of n letters over quaternary alphabet containing the fourth 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, 5, 26, 107, 439, 1663, 6283, 22912, 83507, 298936, 1071564, 3804461, 13542785, 47964338, 170456957, 604293910, 2150604057, 7647190962, 27302834888, 97491596984, 349544018586, 1254140950775, 4517792721381, 16291580640312, 58974333255158, 213744923737227
Offset: 4
A293800
Number of multisets of nonempty words with a total of n letters over quinary alphabet containing the fifth 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, 6, 37, 178, 852, 3767, 16576, 70635, 300433, 1260163, 5287178, 22058377, 92181691, 384663989, 1609206870, 6737170404, 28291573975, 119043337521, 502559706310, 2127305239457, 9035671605064, 38494638887783, 164564824326835, 705742561369640, 3036895648331815
Offset: 5
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 1-10 of 13 results.