OFFSET
0,5
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Wikipedia, Axiom of choice.
FORMULA
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
EXAMPLE
Triangle begins:
1
0 1
0 2 1
1 9 6 1
15 68 48 12 1
222 720 510 150 20 1
3670 9738 6825 2180 360 30 1
68820 159628 110334 36960 6895 735 42 1
Row n = 3 counts the following loop-graphs:
{{1,2},{1,3},{2,3}} {{1},{1,2},{1,3}} {{1},{2},{1,3}} {{1},{2},{3}}
{{1},{1,2},{2,3}} {{1},{2},{2,3}}
{{1},{1,3},{2,3}} {{1},{3},{1,2}}
{{2},{1,2},{1,3}} {{1},{3},{2,3}}
{{2},{1,2},{2,3}} {{2},{3},{1,2}}
{{2},{1,3},{2,3}} {{2},{3},{1,3}}
{{3},{1,2},{1,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
MATHEMATICA
Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}], {n}], Count[#, {_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]], {n, 0, 5}, {k, 0, n}]
PROG
(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)))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 14 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jan 10 2024
EXTENSIONS
a(36) onwards from Andrew Howroyd, Jan 14 2024
STATUS
approved