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A160085
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Number of ordered complementing systems of subsets of {0, 1, ..., n-1} (see A104725).
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0
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0, 1, 1, 1, 3, 1, 5, 1, 13, 3, 5, 1, 33, 1, 5, 5, 75, 1, 33, 1, 33, 5, 5, 1, 261, 3, 5, 13, 33, 1, 61, 1, 541, 5, 5, 5, 375, 1, 5, 5, 261, 1, 61, 1, 33, 33, 5, 1, 2405, 3, 33, 5, 33, 1, 261, 5, 261, 5, 5, 1, 717, 1, 5, 33, 4683, 5, 61, 1, 33, 5, 61, 1, 4549, 1, 5, 33, 33, 5, 61, 1, 2405
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OFFSET
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0,5
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COMMENTS
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Also number of permuted labeled factorizations of n (see the Munagi link for definition and examples)
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LINKS
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FORMULA
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a(0) = 0, a(1) = 1, a(n) = Sum_{j=1..A001222(n)} (Sum_{k=1..j} k!Stirling2(j-1,k-1)), n > 1, where ordfac(n,k) = number of ordered factorizations of n into k factors.
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EXAMPLE
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a(6) = 5: there are 3 complementing systems of subsets of {0,1,2,3,4,5} namely {{0,1,2,3,4,5}}, {{0,1,2},{0,3}} and {{0,1},{0,2,4}} (see A104725). Permuting the components gives 2 additional systems: {{0,3},{0,1,2}} and {{0,2,4},{0,1}}. Thus since {{0,1,2},{0,3}} is a complementing system of subsets of {0,1,2,3,4,5} we have 0 = 0 + 0, 1 = 1 + 0, 2 = 2 + 0, 3 = 0 + 3, 4 = 1 + 3, 5 = 2 + 3.
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MAPLE
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a:=proc(n::integer) local u, r, i, j, k; if n<1 then return 0; elif n=1 then return 1; end if; u:=map(x->x[2], ifactors(n)[2]); r:=add(u[i], i=1..nops(u)); add(add(k!*add((-1)^i*binomial(t, i)*product(binomial(u[j]+t-i-1, u[j]), j=1..nops(u)), i=0..t-1)*stirling2(t-1, k-1), t=k..r), k=1..r); end proc: seq(a(n), n=0..99);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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