A330053 Number of non-isomorphic set-systems of weight n with at least one singleton.

0, 1, 1, 3, 6, 14, 32, 79, 193, 499, 1321, 3626, 10275, 30126, 91062, 284093, 912866, 3018825, 10261530, 35814255, 128197595, 470146011, 1764737593, 6773539331, 26561971320, 106330997834, 434195908353, 1807306022645, 7663255717310, 33079998762373

Offset

0,4

Comments

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.

Links

Jean-François Alcover, Table of n, a(n) for n = 0..50 [using data from A283877 and A306005]

Formula

a(n) = A283877(n) - A306005(n). - Jean-François Alcover, Feb 09 2020

Example

Non-isomorphic representatives of the a(1) = 1 through a(5) = 14 multiset partitions:
  {1}  {1}{2}  {1}{12}    {1}{123}      {1}{1234}
               {1}{23}    {1}{234}      {1}{2345}
               {1}{2}{3}  {1}{2}{12}    {1}{12}{13}
                          {1}{2}{13}    {1}{12}{23}
                          {1}{2}{34}    {1}{12}{34}
                          {1}{2}{3}{4}  {1}{2}{123}
                                        {1}{2}{134}
                                        {1}{2}{345}
                                        {1}{23}{45}
                                        {2}{13}{14}
                                        {1}{2}{3}{12}
                                        {1}{2}{3}{14}
                                        {1}{2}{3}{45}
                                        {1}{2}{3}{4}{5}

Mathematica

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];
A283877 = A@283877;
A306005 = A@306005;
a[n_] := A283877[[n + 1]] - A306005[[n + 1]];
a /@ Range[0, 50] (* Jean-François Alcover, Feb 09 2020 *)

Crossrefs

The complement is counted by A306005.

The multiset partition version is A330058.

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

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

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

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

Sequence in context: A238768 A272362 A182905 * A192678 A114945 A003477

Adjacent sequences: A330050 A330051 A330052 * A330054 A330055 A330056

Keyword

nonn

Author

Gus Wiseman, Nov 30 2019

Status

approved