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A364310 Number T(n,k) of partitions of n into k parts where each block of part i with multiplicity j is marked with a word of length i*j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 4
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 15, 15, 10, 1, 0, 1, 22, 76, 35, 15, 1, 0, 1, 63, 168, 252, 70, 21, 1, 0, 1, 93, 574, 785, 658, 126, 28, 1, 0, 1, 255, 2188, 3066, 2739, 1470, 210, 36, 1, 0, 1, 386, 5490, 18235, 12181, 7857, 2940, 330, 45, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
LINKS
EXAMPLE
T(4,1) = 1: 4abcd.
T(4,2) = 5: 3abc1d, 3abd1c, 3acd1b, 3bcd1a, 22abcd.
T(4,3) = 6: 2ab11cd, 2ac11bd, 2ad11bc, 2bc11ad, 2bd11ac, 2cd11ab.
T(4,4) = 1: 1111abcd.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 3, 1;
0, 1, 5, 6, 1;
0, 1, 15, 15, 10, 1;
0, 1, 22, 76, 35, 15, 1;
0, 1, 63, 168, 252, 70, 21, 1;
0, 1, 93, 574, 785, 658, 126, 28, 1;
0, 1, 255, 2188, 3066, 2739, 1470, 210, 36, 1;
...
MAPLE
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*x^j*binomial(n, i*j), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);
MATHEMATICA
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
Sum[b[n - i*j, i - 1]*x^j*Binomial[n, i*j], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 18 2023, after Alois P. Heinz *)
CROSSREFS
Columns k=0-1 give: A000007, A057427.
Row sums give A178682.
T(n,n) gives A000012.
T(n+1,n) gives A000217.
T(n+2,n) gives A000332(n+3).
Cf. A364285.
Sequence in context: A327618 A121314 A119271 * A323222 A125104 A098157
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 18 2023
STATUS
approved

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Last modified May 29 00:29 EDT 2024. Contains 372921 sequences. (Running on oeis4.)