A368596 - OEIS

%I #15 Feb 24 2024 11:03:16

%S 0,0,0,3,66,1380,31460,800625,22758918,718821852,25057509036,

%T 957657379437,39878893266795,1799220308202603,87502582432459584,

%U 4566246347310609247,254625879822078742956,15115640124974801925030,952050565540607423524658,63425827673509972464868323

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

%C 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 Andrew Howroyd, <a href="/A368596/b368596.txt">Table of n, a(n) for n = 0..200</a>

%e The a(3) = 3 set-systems:

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

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

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

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

%Y The version without the choice condition is A014068, covering A368597.

%Y The complement appears to be A333331.

%Y For covering pairs we have A367868.

%Y Allowing edges of any positive size gives A368600, any length A367903.

%Y The covering case is A368730.

%Y The unlabeled version is A368835.

%Y A000085 counts set partitions into singletons or pairs.

%Y A006125 counts graphs, unlabeled A000088.

%Y A058891 counts set-systems (without singletons A016031), unlabeled A000612.

%Y A100861 counts set partitions into singletons or pairs by number of pairs.

%Y A111924 counts set partitions into singletons or pairs by length.

%Y A322661 counts covering half-loop-graphs, connected A062740.

%Y A369141 counts non-choosable loop-graphs, covering A369142.

%Y A369146 counts unlabeled non-choosable loop-graphs, covering A369147.

%Y Cf. A000272, A000666, A057500, A129271, A133686, A367769, A367863, A367867, A367869, A367901, A367907, A368097, A369199.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 04 2024

%E Terms a(7) and beyond from _Andrew Howroyd_, Jan 10 2024