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
Examples
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; ...
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)|.
Crossrefs
Programs
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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
Formula
Extensions
Offset changed and terms a(44) and beyond from Andrew Howroyd, Apr 25 2020