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A213427
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Number of ways of refining the partition n^1 to get 1^n.
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39
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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MAPLE
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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]):
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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