cp's OEIS Frontend

A368596 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.

%I A368596 #15 Feb 24 2024 11:03:16
%S A368596 0,0,0,3,66,1380,31460,800625,22758918,718821852,25057509036,
%T A368596 957657379437,39878893266795,1799220308202603,87502582432459584,
%U A368596 4566246347310609247,254625879822078742956,15115640124974801925030,952050565540607423524658,63425827673509972464868323
%N A368596 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 A368596 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 A368596 Andrew Howroyd, <a href="/A368596/b368596.txt">Table of n, a(n) for n = 0..200</a>
%e A368596 The a(3) = 3 set-systems:
%e A368596   {{1},{2},{1,2}}
%e A368596   {{1},{3},{1,3}}
%e A368596   {{2},{3},{2,3}}
%t A368596 Table[Length[Select[Subsets[Subsets[Range[n],{1,2}], {n}],Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]
%Y A368596 The version without the choice condition is A014068, covering A368597.
%Y A368596 The complement appears to be A333331.
%Y A368596 For covering pairs we have A367868.
%Y A368596 Allowing edges of any positive size gives A368600, any length A367903.
%Y A368596 The covering case is A368730.
%Y A368596 The unlabeled version is A368835.
%Y A368596 A000085 counts set partitions into singletons or pairs.
%Y A368596 A006125 counts graphs, unlabeled A000088.
%Y A368596 A058891 counts set-systems (without singletons A016031), unlabeled A000612.
%Y A368596 A100861 counts set partitions into singletons or pairs by number of pairs.
%Y A368596 A111924 counts set partitions into singletons or pairs by length.
%Y A368596 A322661 counts covering half-loop-graphs, connected A062740.
%Y A368596 A369141 counts non-choosable loop-graphs, covering A369142.
%Y A368596 A369146 counts unlabeled non-choosable loop-graphs, covering A369147.
%Y A368596 Cf. A000272, A000666, A057500, A129271, A133686, A367769, A367863, A367867, A367869, A367901, A367907, A368097, A369199.
%K A368596 nonn
%O A368596 0,4
%A A368596 _Gus Wiseman_, Jan 04 2024
%E A368596 Terms a(7) and beyond from _Andrew Howroyd_, Jan 10 2024