A238967 Maximal size of an antichain in canonical order.
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, 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, 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
Offset: 0
Examples
Triangle T(n,k) begins: 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; ...
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.
Programs
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Maple
with(numtheory): f:= n-> (m-> add(`if`(bigomega(d)=m, 1, 0), d=divisors(n)))(iquo(bigomega(n), 2)): b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x-> [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]): T:= n-> map(x-> f(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]: seq(T(n), n=0..9); # Alois P. Heinz, Mar 26 2020 -
PARI
\\ here b(n) is A096825. b(n)={my(h=bigomega(n)\2); sumdiv(n, d, bigomega(d)==h)} N(sig)={prod(k=1, #sig, prime(k)^sig[k])} Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))} { for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 25 2020
Formula
Extensions
Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 25 2020