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%I #23 May 02 2016 12:28:41
%S 1,1,2,6,18,74,314,1614,8650,52794,337410,2373822,17327770,136539154,
%T 1115206818,9671306438,86529147794,816066328602,7904640819682,
%U 80089651530566,832008919174434,8983256694817802,99219778649809162,1134999470682805134,13241030890523397154
%N Number of ways of refining the partition n^1 to get 1^n.
%C 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.
%C Then a(n) is the total number of paths in the augmented poset of any length from n^1 to 1^n.
%H Alois P. Heinz, <a href="/A213427/b213427.txt">Table of n, a(n) for n = 1..40</a>
%H Olivier Gérard, <a href="/A002846/a002846.png">The ranked posets L(2),...,L(8)</a>
%p b:= proc(l) option remember; local i, j, n, t; n:=nops(l);
%p `if`(n<2, {[0]}, `if`(l[-1]=0, b(subsop(n=NULL, l)), {l,
%p seq(`if`(l[i]=0, {}[], {seq(b([seq(l[t]-`if`(t=1, l[t],
%p `if`(t=i, 1, `if`(t=j and t=i-j, -2, `if`(t=j or t=i-j,
%p -1, 0)))), t=1..n)])[], j=1..i/2)}[]), i=2..n)}))
%p end:
%p p:= proc(l) option remember;
%p `if`(nops(l)=1, 1, add(p(x), x=b(l) minus {l}))
%p end:
%p a:= n-> p([0$(n-1), 1]):
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Jun 12 2012
%Y Cf. A002846, A213242, A213385.
%K nonn
%O 1,3
%A _N. J. A. Sloane_, Jun 11 2012
%E More terms from _Alois P. Heinz_, Jun 11 2012
%E Edited by _Alois P. Heinz_ at the suggestion of _Gus Wiseman_, May 02 2016
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