A238967 - OEIS

%I #14 Mar 26 2020 18:47:37

%S 1,1,1,2,1,2,3,1,2,3,4,6,1,2,3,4,5,7,10,1,2,3,4,4,6,8,7,10,14,20,1,2,

%T 3,4,4,6,8,7,8,11,15,13,18,25,35,1,2,3,4,4,6,8,5,8,9,12,16,10,14,16,

%U 22,30,19,26,36,50,70,1,2,3,4,4,6,8,5,8,9,12,16,9,11,15,17,23,31,12,19,26,22,30,41,56,35,48,66,91,126

%N Maximal size of an antichain in canonical order.

%H Andrew Howroyd, <a href="/A238967/b238967.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.

%F T(n,k) = A096825(A063008(n,k)). - _Andrew Howroyd_, Mar 25 2020

%e Triangle T(n,k) begins:

%e   1;

%e   1;

%e   1, 2;

%e   1, 2, 3;

%e   1, 2, 3, 4, 6;

%e   1, 2, 3, 4, 5, 7, 10;

%e   1, 2, 3, 4, 4, 6,  8, 7, 10, 14, 20;

%e   ...

%p with(numtheory):

%p f:= n-> (m-> add(`if`(bigomega(d)=m, 1, 0),

%p      d=divisors(n)))(iquo(bigomega(n), 2)):

%p b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->

%p     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):

%p T:= n-> map(x-> f(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:

%p seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 26 2020

%o (PARI) \\ here b(n) is A096825.

%o b(n)={my(h=bigomega(n)\2); sumdiv(n, d, bigomega(d)==h)}

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

%o Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}

%o { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 25 2020

%Y Cf. A238954 in canonical order.

%Y Cf. A063008, A096825.

%K nonn,tabf

%O 0,4

%A _Sung-Hyuk Cha_, Mar 07 2014

%E Offset changed and terms a(50) and beyond from _Andrew Howroyd_, Mar 25 2020