A213427 Number of ways of refining the partition n^1 to get 1^n.

1, 1, 2, 6, 18, 74, 314, 1614, 8650, 52794, 337410, 2373822, 17327770, 136539154, 1115206818, 9671306438, 86529147794, 816066328602, 7904640819682, 80089651530566, 832008919174434, 8983256694817802, 99219778649809162, 1134999470682805134, 13241030890523397154

Offset

1,3

Comments

Consider the ranked poset L(n) of partitions defined in A002846. Add additional edges from each partition to any other partition that is a refinement of it. In L(5), for example, we add edges from 5^1 to 31^2, 2^21, 21^3 and 1^5, from 41 to 21^3 and 1^5, and so on.

Then a(n) is the total number of paths in the augmented poset of any length from n^1 to 1^n.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..40

Olivier Gérard, The ranked posets L(2),...,L(8)

Maple

b:= proc(l) option remember; local i, j, n, t; n:=nops(l);
      `if`(n<2, {[0]}, `if`(l[-1]=0, b(subsop(n=NULL, l)), {l,
      seq(`if`(l[i]=0, {}[], {seq(b([seq(l[t]-`if`(t=1, l[t],
      `if`(t=i, 1, `if`(t=j and t=i-j, -2, `if`(t=j or t=i-j,
      -1, 0)))), t=1..n)])[], j=1..i/2)}[]), i=2..n)}))
    end:
p:= proc(l) option remember;
      `if`(nops(l)=1, 1, add(p(x), x=b(l) minus {l}))
    end:
a:= n-> p([0$(n-1), 1]):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 12 2012

Crossrefs

Cf. A002846, A213242, A213385.

Sequence in context: A022491 A004395 A277862 * A006388 A007116 A280763

Adjacent sequences: A213424 A213425 A213426 * A213428 A213429 A213430

Keyword

nonn

Author

N. J. A. Sloane, Jun 11 2012

Extensions

More terms from Alois P. Heinz, Jun 11 2012

Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016

Status

approved