cp's OEIS Frontend

A368597 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.

%I A368597 #15 Feb 24 2024 11:03:20
%S A368597 1,1,3,17,150,1803,27364,501015,10736010,263461265,7283725704,
%T A368597 223967628066,7581128184175,280103206674480,11216492736563655,
%U A368597 483875783716549277,22371631078155742764,1103548801569848115255,57849356643299101021960,3211439288584038922502820
%N A368597 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.
%C A368597 It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.
%H A368597 Andrew Howroyd, <a href="/A368597/b368597.txt">Table of n, a(n) for n = 0..200</a>
%H A368597 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphLoop.html">Graph Loop</a>.
%F A368597 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n). - _Andrew Howroyd_, Jan 06 2024
%e A368597 The a(0) = 1 through a(3) = 17 set-systems:
%e A368597   {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
%e A368597              {{1},{1,2}}  {{1},{2},{1,3}}
%e A368597              {{2},{1,2}}  {{1},{2},{2,3}}
%e A368597                           {{1},{3},{1,2}}
%e A368597                           {{1},{3},{2,3}}
%e A368597                           {{2},{3},{1,2}}
%e A368597                           {{2},{3},{1,3}}
%e A368597                           {{1},{1,2},{1,3}}
%e A368597                           {{1},{1,2},{2,3}}
%e A368597                           {{1},{1,3},{2,3}}
%e A368597                           {{2},{1,2},{1,3}}
%e A368597                           {{2},{1,2},{2,3}}
%e A368597                           {{2},{1,3},{2,3}}
%e A368597                           {{3},{1,2},{1,3}}
%e A368597                           {{3},{1,2},{2,3}}
%e A368597                           {{3},{1,3},{2,3}}
%e A368597                           {{1,2},{1,3},{2,3}}
%t A368597 Table[Length[Select[Subsets[Subsets[Range[n],{1,2}], {n}],Union@@#==Range[n]&]],{n,0,5}]
%o A368597 (PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n)) \\ _Andrew Howroyd_, Jan 06 2024
%Y A368597 This is the covering case of A014068.
%Y A368597 Allowing edges of any positive size gives A054780, covering case of A136556.
%Y A368597 Allowing any number of edges gives A322661, connected A062740.
%Y A368597 The case of just pairs is A367863, covering case of A116508.
%Y A368597 The unlabeled version is A368599.
%Y A368597 The version contradicting strict AOC is A368730.
%Y A368597 The connected case is A368951.
%Y A368597 A000085 counts set partitions into singletons or pairs.
%Y A368597 A006129 counts covering graphs, connected A001187.
%Y A368597 A058891 counts set-systems, unlabeled A000612.
%Y A368597 A100861 counts set partitions into singletons or pairs by number of pairs.
%Y A368597 A111924 counts set partitions into singletons or pairs by length.
%Y A368597 Cf. A000272, A000666, A057500, A066383, A333331, A367869, A368596, A368600, A368928, A368951, A369199.
%K A368597 nonn
%O A368597 0,3
%A A368597 _Gus Wiseman_, Jan 04 2024
%E A368597 Terms a(7) and beyond from _Andrew Howroyd_, Jan 06 2024