cp's OEIS Frontend

A369141 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).

%I A369141 #9 Feb 02 2024 18:34:44
%S A369141 0,0,1,25,710,29394,2051522,267690539,68705230758,35184059906570,
%T A369141 36028789310419722,73786976083150073999,302231454897259573627852,
%U A369141 2475880078570549574773324062,40564819207303333310731978895956,1329227995784915872613854321228773937
%N A369141 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 A369141 Also labeled loop-graphs having at least one connected component containing more edges than vertices.
%H A369141 Andrew Howroyd, <a href="/A369141/b369141.txt">Table of n, a(n) for n = 0..50</a>
%F A369141 Binomial transform of A369142.
%F A369141 a(n) = A006125(n + 1) - A368927(n). - _Andrew Howroyd_, Feb 02 2024
%e A369141 The a(0) = 0 through a(3) = 25 loop-graphs (loops shown as singletons):
%e A369141   .  .  {{1},{2},{1,2}}  {{1},{2},{1,2}}
%e A369141                          {{1},{3},{1,3}}
%e A369141                          {{2},{3},{2,3}}
%e A369141                          {{1},{2},{3},{1,2}}
%e A369141                          {{1},{2},{3},{1,3}}
%e A369141                          {{1},{2},{3},{2,3}}
%e A369141                          {{1},{2},{1,2},{1,3}}
%e A369141                          {{1},{2},{1,2},{2,3}}
%e A369141                          {{1},{2},{1,3},{2,3}}
%e A369141                          {{1},{3},{1,2},{1,3}}
%e A369141                          {{1},{3},{1,2},{2,3}}
%e A369141                          {{1},{3},{1,3},{2,3}}
%e A369141                          {{2},{3},{1,2},{1,3}}
%e A369141                          {{2},{3},{1,2},{2,3}}
%e A369141                          {{2},{3},{1,3},{2,3}}
%e A369141                          {{1},{1,2},{1,3},{2,3}}
%e A369141                          {{2},{1,2},{1,3},{2,3}}
%e A369141                          {{3},{1,2},{1,3},{2,3}}
%e A369141                          {{1},{2},{3},{1,2},{1,3}}
%e A369141                          {{1},{2},{3},{1,2},{2,3}}
%e A369141                          {{1},{2},{3},{1,3},{2,3}}
%e A369141                          {{1},{2},{1,2},{1,3},{2,3}}
%e A369141                          {{1},{3},{1,2},{1,3},{2,3}}
%e A369141                          {{2},{3},{1,2},{1,3},{2,3}}
%e A369141                          {{1},{2},{3},{1,2},{1,3},{2,3}}
%t A369141 Table[Length[Select[Subsets[Subsets[Range[n], {1,2}]],Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]
%Y A369141 Without the choice condition we have A006125, unlabeled A000088.
%Y A369141 The case of a unique choice is A088957, unlabeled A087803.
%Y A369141 The case without loops is A367867, covering A367868.
%Y A369141 For edges of any positive size we have A367903, complement A367902.
%Y A369141 For exactly n edges we have A368596, complement A333331 (maybe).
%Y A369141 The complement is counted by A368927, covering A369140.
%Y A369141 The covering case is A369142.
%Y A369141 For n edges and no loops we have A369143, covering A369144.
%Y A369141 The unlabeled version is A369146 (covering A369147), complement A369145.
%Y A369141 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A369141 A006129 counts covering graphs, unlabeled A002494.
%Y A369141 A054548 counts graphs covering n vertices with k edges, with loops A369199.
%Y A369141 A133686 counts choosable graphs, covering A367869.
%Y A369141 A322661 counts labeled covering loop-graphs, unlabeled A322700.
%Y A369141 Cf. A000272, A000666, A006649, A062740, A116508, A137916, A368924, A368984, A369194.
%K A369141 nonn
%O A369141 0,4
%A A369141 _Gus Wiseman_, Jan 20 2024
%E A369141 a(6) onwards from _Andrew Howroyd_, Feb 02 2024