%I #14 Jan 14 2024 16:10:56

%S 1,0,1,0,2,1,1,9,6,1,15,68,48,12,1,222,720,510,150,20,1,3670,9738,

%T 6825,2180,360,30,1,68820,159628,110334,36960,6895,735,42,1,1456875,

%U 3067320,2090760,721560,145530,17976,1344,56,1,34506640,67512798,45422928,15989232,3402756,463680,40908,2268,72,1

%N Triangle read by rows where T(n,k) is the number of labeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different vertex from each edge.

%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="/A368924/b368924.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.

%F E.g.f.: A(x,y) = exp(-log(1-T(x))/2 - T(x)/2 + y*T(x) - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - _Andrew Howroyd_, Jan 14 2024

%e Triangle begins:

%e 1

%e 0 1

%e 0 2 1

%e 1 9 6 1

%e 15 68 48 12 1

%e 222 720 510 150 20 1

%e 3670 9738 6825 2180 360 30 1

%e 68820 159628 110334 36960 6895 735 42 1

%e Row n = 3 counts the following loop-graphs:

%e {{1,2},{1,3},{2,3}} {{1},{1,2},{1,3}} {{1},{2},{1,3}} {{1},{2},{3}}

%e {{1},{1,2},{2,3}} {{1},{2},{2,3}}

%e {{1},{1,3},{2,3}} {{1},{3},{1,2}}

%e {{2},{1,2},{1,3}} {{1},{3},{2,3}}

%e {{2},{1,2},{2,3}} {{2},{3},{1,2}}

%e {{2},{1,3},{2,3}} {{2},{3},{1,3}}

%e {{3},{1,2},{1,3}}

%e {{3},{1,2},{2,3}}

%e {{3},{1,3},{2,3}}

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

%o (PARI) T(n)={my(t=-lambertw(-x + O(x*x^n))); [Vecrev(p) | p <- Vec(serlaplace(exp(-log(1-t)/2 - t/2 + t*y - t^2/4)))]}

%o { my(A=T(8)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 14 2024

%Y Column k = n-1 is A002378.

%Y The case of a unique choice is A061356, row sums A000272.

%Y Column k = 0 is A137916, unlabeled version A137917.

%Y Row sums appear to be A333331.

%Y The complement has row sums A368596, covering case A368730.

%Y The unlabeled version is A368926.

%Y Without the choice condition we have A368928, A116508, A367863, A368597.

%Y A000085, A100861, A111924 count set partitions into singletons or pairs.

%Y A006125 counts graphs, unlabeled A000088.

%Y A006129 counts covering graphs, unlabeled A002494.

%Y A014068 counts loop-graphs, unlabeled A000666.

%Y Cf. A000169, A057500, A062740, A129271, A133686, A322661, A367869, A367902, A368601, A368835, A368836, A368927.

%K nonn,tabl

%O 0,5

%A _Gus Wiseman_, Jan 10 2024

%E a(36) onwards from _Andrew Howroyd_, Jan 14 2024