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Number of non-isomorphic set-systems of weight n with at least one singleton.
6

%I #18 Feb 10 2020 06:52:48

%S 0,1,1,3,6,14,32,79,193,499,1321,3626,10275,30126,91062,284093,912866,

%T 3018825,10261530,35814255,128197595,470146011,1764737593,6773539331,

%U 26561971320,106330997834,434195908353,1807306022645,7663255717310,33079998762373

%N Number of non-isomorphic set-systems of weight n with at least one singleton.

%C A set-system is a finite set of finite nonempty sets of positive integers. An singleton is an edge of size 1. The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

%H Jean-François Alcover, <a href="/A330053/b330053.txt">Table of n, a(n) for n = 0..50</a> [using data from A283877 and A306005]

%F a(n) = A283877(n) - A306005(n). - _Jean-François Alcover_, Feb 09 2020

%e Non-isomorphic representatives of the a(1) = 1 through a(5) = 14 multiset partitions:

%e {1} {1}{2} {1}{12} {1}{123} {1}{1234}

%e {1}{23} {1}{234} {1}{2345}

%e {1}{2}{3} {1}{2}{12} {1}{12}{13}

%e {1}{2}{13} {1}{12}{23}

%e {1}{2}{34} {1}{12}{34}

%e {1}{2}{3}{4} {1}{2}{123}

%e {1}{2}{134}

%e {1}{2}{345}

%e {1}{23}{45}

%e {2}{13}{14}

%e {1}{2}{3}{12}

%e {1}{2}{3}{14}

%e {1}{2}{3}{45}

%e {1}{2}{3}{4}{5}

%t A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];

%t A283877 = A@283877;

%t A306005 = A@306005;

%t a[n_] := A283877[[n + 1]] - A306005[[n + 1]];

%t a /@ Range[0, 50] (* _Jean-François Alcover_, Feb 09 2020 *)

%Y The complement is counted by A306005.

%Y The multiset partition version is A330058.

%Y Non-isomorphic set-systems with at least one endpoint are A330052.

%Y Non-isomorphic set-systems counted by vertices are A000612.

%Y Non-isomorphic set-systems counted by weight are A283877.

%Y Cf. A007716, A055621, A302545, A317533, A317794, A319559, A320665, A330055, A330056, A330057.

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 30 2019