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

Author

N. J. A. Sloane, Jun 11 2012

Keywords

• nonn

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)

Crossrefs

Cf. A002846, A213242, A213385.

Programs

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

Extensions

More terms from Alois P. Heinz, Jun 11 2012

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