cp's OEIS Frontend

A368730 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges, such that it is not possible to choose a different element from each.

%I A368730 #15 Jan 10 2024 15:01:10
%S A368730 0,0,0,0,6,180,4560,117600,3234588,96119982,3092585310,107542211535,
%T A368730 4029055302855,162040513972623,6970457656110039,319598974394563500,
%U A368730 15568332397812799920,803271954062642638830,43778508937914677872788,2513783434620146896920843
%N A368730 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges, such that it is not possible to choose a different element from each.
%C A368730 The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
%H A368730 Andrew Howroyd, <a href="/A368730/b368730.txt">Table of n, a(n) for n = 0..200</a>
%H A368730 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphLoop.html">Graph Loop</a>.
%H A368730 ProofWiki, <a href="https://proofwiki.org/wiki/Definition:Loop-Graph">Definition:Loop-Graph</a>
%F A368730 a(n) = A368596(n) + A368597(n) - A014068(n). - _Andrew Howroyd_, Jan 10 2024
%e A368730 The a(4) = 6 set-systems:
%e A368730   {{1},{2},{1,2},{3,4}}
%e A368730   {{1},{3},{1,3},{2,4}}
%e A368730   {{1},{4},{1,4},{2,3}}
%e A368730   {{2},{3},{1,4},{2,3}}
%e A368730   {{2},{4},{1,3},{2,4}}
%e A368730   {{3},{4},{1,2},{3,4}}
%t A368730 Table[Length[Select[Subsets[Subsets[Range[n],{1,2}], {n}],Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]
%Y A368730 The case of a unique choice appears to be A000272.
%Y A368730 The version without the choice condition is A368597, non-covering A014068.
%Y A368730 The complement appears to be A333331.
%Y A368730 The non-covering case is A368596, allowing edges of any size A368600.
%Y A368730 Allowing any number of edges of any size gives A367903, ranks A367907.
%Y A368730 Allowing any number of non-singletons gives A367868, non-covering A367867.
%Y A368730 A000085 counts set partitions into singletons or pairs.
%Y A368730 A006125 counts graphs, unlabeled A000088.
%Y A368730 A006129 counts covering graphs, unlabeled A002494.
%Y A368730 A100861 counts set partitions into singletons or pairs by number of pairs.
%Y A368730 A111924 counts set partitions into singletons or pairs by length.
%Y A368730 A322661 counts labeled covering half-loop-graphs, connected A062740.
%Y A368730 Cf. A000666, A116508, A133686, A367863, A367869, A367901, A368532.
%K A368730 nonn
%O A368730 0,5
%A A368730 _Gus Wiseman_, Jan 04 2024
%E A368730 Terms a(7) and beyond from _Andrew Howroyd_, Jan 10 2024