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A213427 Number of ways of refining the partition n^1 to get 1^n. 39
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A022491 A004395 A277862 * A006388 A007116 A280763
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

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Last modified May 20 05:07 EDT 2024. Contains 372703 sequences. (Running on oeis4.)