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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 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

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