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A055621
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Number of covers of an unlabeled n-set.
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108
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OFFSET
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0,3
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 78 (2.3.39)
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LINKS
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Eric Weisstein's World of Mathematics, Cover
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FORMULA
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EXAMPLE
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There are 4 nonisomorphic covers of {1,2}, namely {{1},{2}}, {{1,2}}, {{1},{1,2}} and {{1},{2},{1,2}}.
Non-isomorphic representatives of the a(3) = 34 covers:
{123} {1}{23} {1}{2}{3} {1}{2}{3}{23}
{13}{23} {1}{3}{23} {1}{2}{13}{23}
{3}{123} {2}{13}{23} {1}{2}{3}{123}
{23}{123} {2}{3}{123} {2}{3}{13}{23}
{3}{13}{23} {1}{3}{23}{123}
{12}{13}{23} {2}{3}{23}{123}
{1}{23}{123} {3}{12}{13}{23}
{3}{23}{123} {2}{13}{23}{123}
{13}{23}{123} {3}{13}{23}{123}
{12}{13}{23}{123}
.
{1}{2}{3}{13}{23} {1}{2}{3}{12}{13}{23} {1}{2}{3}{12}{13}{23}{123}
{1}{2}{3}{23}{123} {1}{2}{3}{13}{23}{123}
{2}{3}{12}{13}{23} {2}{3}{12}{13}{23}{123}
{1}{2}{13}{23}{123}
{2}{3}{13}{23}{123}
{3}{12}{13}{23}{123}
(End)
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MAPLE
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b:= proc(n, i, l) `if`(n=0, 2^(w-> add(mul(2^igcd(t, l[h]),
h=1..nops(l)), t=1..w)/w)(ilcm(l[])), `if`(i<1, 0,
add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i)))
end:
a:= n-> `if`(n=0, 2, b(n$2, [])-b(n-1$2, []))/2:
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MATHEMATICA
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b[n_, i_, l_] := b[n, i, l] = If[n==0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][If[l=={}, 1, LCM@@l]], If[i<1, 0, Sum[b[n-i*j, i-1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];
a[n_] := If[n==0, 2, b[n, n, {}] - b[n-1, n-1, {}]]/2;
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CROSSREFS
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Unlabeled set-systems are A000612 (partial sums).
The version with empty edges allowed is A003181.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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More terms from David Moews (dmoews(AT)xraysgi.ims.uconn.edu) Jul 04 2002
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STATUS
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approved
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