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.
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
Offset: 0
Examples
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; ...
Links
- Alois P. Heinz, Rows n = 0..150, flattened
Crossrefs
Programs
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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 *)