A238962 - OEIS

%I #18 Aug 30 2020 20:18:10

%S 1,1,2,3,4,8,13,8,20,26,44,75,16,48,76,132,176,308,541,32,112,208,252,

%T 368,604,818,1076,1460,2612,4683,64,256,544,768,976,1888,2316,3172,

%U 3408,5740,7880,10404,14300,25988,47293,128,576,1376,2208,2568,2496,5536,7968

%N Number of perfect partitions in graded colexicographic order.

%H Andrew Howroyd, <a href="/A238962/b238962.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)

%H S.-H. Cha, E. G. DuCasse, and L. V. Quintas, <a href="http://arxiv.org/abs/1405.5283">Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures</a>, arXiv:1405.5283 [math.NT], 2014, Table A.1 entry |P^T(s)|.

%F T(n,k) = A074206(A036035(n,k)). - _Andrew Howroyd_, Apr 25 2020

%e Triangle T(n,k) begins:

%e    1;

%e    1;

%e    2,   3;

%e    4,   8,  13;

%e    8,  20,  26,  44,  75;

%e   16,  48,  76, 132, 176, 308, 541;

%e   32, 112, 208, 252, 368, 604, 818, 1076, 1460, 2612, 4683;

%e   ...

%o (PARI) \\ here b(n) is A074206.

%o N(sig)={prod(k=1, #sig, prime(k)^sig[k])}

%o 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))))}

%o Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}

%o { for(n=0, 6, print(Row(n))) } \\ _Andrew Howroyd_, Aug 30 2020

%Y Row sums are A035341.

%Y Cf. A002033 in graded colexicographic order.

%Y Cf. A036035, A074206, A238975.

%K nonn,tabf

%O 0,3

%A _Sung-Hyuk Cha_, Mar 07 2014

%E Offset changed and terms a(44) and beyond from _Andrew Howroyd_, Apr 25 2020