A369141 - OEIS

%I #9 Feb 02 2024 18:34:44

%S 0,0,1,25,710,29394,2051522,267690539,68705230758,35184059906570,

%T 36028789310419722,73786976083150073999,302231454897259573627852,

%U 2475880078570549574773324062,40564819207303333310731978895956,1329227995784915872613854321228773937

%N Number of labeled loop-graphs covering a subset of {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

%C Also labeled loop-graphs having at least one connected component containing more edges than vertices.

%H Andrew Howroyd, <a href="/A369141/b369141.txt">Table of n, a(n) for n = 0..50</a>

%F Binomial transform of A369142.

%F a(n) = A006125(n + 1) - A368927(n). - _Andrew Howroyd_, Feb 02 2024

%e The a(0) = 0 through a(3) = 25 loop-graphs (loops shown as singletons):

%e   .  .  {{1},{2},{1,2}}  {{1},{2},{1,2}}

%e                          {{1},{3},{1,3}}

%e                          {{2},{3},{2,3}}

%e                          {{1},{2},{3},{1,2}}

%e                          {{1},{2},{3},{1,3}}

%e                          {{1},{2},{3},{2,3}}

%e                          {{1},{2},{1,2},{1,3}}

%e                          {{1},{2},{1,2},{2,3}}

%e                          {{1},{2},{1,3},{2,3}}

%e                          {{1},{3},{1,2},{1,3}}

%e                          {{1},{3},{1,2},{2,3}}

%e                          {{1},{3},{1,3},{2,3}}

%e                          {{2},{3},{1,2},{1,3}}

%e                          {{2},{3},{1,2},{2,3}}

%e                          {{2},{3},{1,3},{2,3}}

%e                          {{1},{1,2},{1,3},{2,3}}

%e                          {{2},{1,2},{1,3},{2,3}}

%e                          {{3},{1,2},{1,3},{2,3}}

%e                          {{1},{2},{3},{1,2},{1,3}}

%e                          {{1},{2},{3},{1,2},{2,3}}

%e                          {{1},{2},{3},{1,3},{2,3}}

%e                          {{1},{2},{1,2},{1,3},{2,3}}

%e                          {{1},{3},{1,2},{1,3},{2,3}}

%e                          {{2},{3},{1,2},{1,3},{2,3}}

%e                          {{1},{2},{3},{1,2},{1,3},{2,3}}

%t Table[Length[Select[Subsets[Subsets[Range[n], {1,2}]],Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

%Y Without the choice condition we have A006125, unlabeled A000088.

%Y The case of a unique choice is A088957, unlabeled A087803.

%Y The case without loops is A367867, covering A367868.

%Y For edges of any positive size we have A367903, complement A367902.

%Y For exactly n edges we have A368596, complement A333331 (maybe).

%Y The complement is counted by A368927, covering A369140.

%Y The covering case is A369142.

%Y For n edges and no loops we have A369143, covering A369144.

%Y The unlabeled version is A369146 (covering A369147), complement A369145.

%Y A000085, A100861, A111924 count set partitions into singletons or pairs.

%Y A006129 counts covering graphs, unlabeled A002494.

%Y A054548 counts graphs covering n vertices with k edges, with loops A369199.

%Y A133686 counts choosable graphs, covering A367869.

%Y A322661 counts labeled covering loop-graphs, unlabeled A322700.

%Y Cf. A000272, A000666, A006649, A062740, A116508, A137916, A368924, A368984, A369194.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 20 2024

%E a(6) onwards from _Andrew Howroyd_, Feb 02 2024