cp's OEIS Frontend

A238967 Maximal size of an antichain in canonical order.

%I A238967 #14 Mar 26 2020 18:47:37
%S A238967 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 A238967 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 A238967 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 A238967 Maximal size of an antichain in canonical order.
%H A238967 Andrew Howroyd, <a href="/A238967/b238967.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H A238967 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 A238967 T(n,k) = A096825(A063008(n,k)). - _Andrew Howroyd_, Mar 25 2020
%e A238967 Triangle T(n,k) begins:
%e A238967   1;
%e A238967   1;
%e A238967   1, 2;
%e A238967   1, 2, 3;
%e A238967   1, 2, 3, 4, 6;
%e A238967   1, 2, 3, 4, 5, 7, 10;
%e A238967   1, 2, 3, 4, 4, 6,  8, 7, 10, 14, 20;
%e A238967   ...
%p A238967 with(numtheory):
%p A238967 f:= n-> (m-> add(`if`(bigomega(d)=m, 1, 0),
%p A238967      d=divisors(n)))(iquo(bigomega(n), 2)):
%p A238967 b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p A238967     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A238967 T:= n-> map(x-> f(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
%p A238967 seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 26 2020
%o A238967 (PARI) \\ here b(n) is A096825.
%o A238967 b(n)={my(h=bigomega(n)\2); sumdiv(n, d, bigomega(d)==h)}
%o A238967 N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
%o A238967 Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
%o A238967 { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 25 2020
%Y A238967 Cf. A238954 in canonical order.
%Y A238967 Cf. A063008, A096825.
%K A238967 nonn,tabf
%O A238967 0,4
%A A238967 _Sung-Hyuk Cha_, Mar 07 2014
%E A238967 Offset changed and terms a(50) and beyond from _Andrew Howroyd_, Mar 25 2020