A238962 Number of perfect partitions in graded colexicographic order.

1, 1, 2, 3, 4, 8, 13, 8, 20, 26, 44, 75, 16, 48, 76, 132, 176, 308, 541, 32, 112, 208, 252, 368, 604, 818, 1076, 1460, 2612, 4683, 64, 256, 544, 768, 976, 1888, 2316, 3172, 3408, 5740, 7880, 10404, 14300, 25988, 47293, 128, 576, 1376, 2208, 2568, 2496, 5536, 7968

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)

S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014, Table A.1 entry |P^T(s)|.

Formula

T(n,k) = A074206(A036035(n,k)). - Andrew Howroyd, Apr 25 2020

Example

Triangle T(n,k) begins:
   1;
   1;
   2,   3;
   4,   8,  13;
   8,  20,  26,  44,  75;
  16,  48,  76, 132, 176, 308, 541;
  32, 112, 208, 252, 368, 604, 818, 1076, 1460, 2612, 4683;
  ...

Prog

(PARI) \\ here b(n) is A074206.
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
b(n)={if(!n, 0, my(sig=factor(n)[, 2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r, k)*(-1)^(r-k))))}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 6, print(Row(n))) } \\ Andrew Howroyd, Aug 30 2020

Crossrefs

Row sums are A035341.

Cf. A002033 in graded colexicographic order.

Cf. A036035, A074206, A238975.

Sequence in context: A226947 A272615 A356188 * A238975 A098348 A131420

Adjacent sequences: A238959 A238960 A238961 * A238963 A238964 A238965

Keyword

nonn,tabf

Author

Sung-Hyuk Cha, Mar 07 2014

Extensions

Offset changed and terms a(44) and beyond from Andrew Howroyd, Apr 25 2020

Status

approved