A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

0, 1, 2, 4, 11, 25, 75, 206, 650, 2049, 6895

Offset

0,3

Comments

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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

Table of n, a(n) for n=0..10.

Wikipedia, Axiom of choice.

Example

Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{2},{1,2}}  {{1,2,3,3}}
                               {{1,2,3,4}}
                               {{1},{1,2,2}}
                               {{1,2},{1,2}}
                               {{1,2},{2,2}}
                               {{1,3},{2,3}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s, Flatten[MapIndexed[Table[#2, {#1}]&, #]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i, p[[i]]}, {i, Length[p]}])], {p, Permutations[Union@@m]}]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[brute /@ Select[mpm[n], Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]], {n, 0, 6}]

Crossrefs

The case of labeled graphs is A129271, connected case of A133686.

The complement for labeled graphs is A140638, connected case of A367867.

This is the connected case of A368098, ranks A368100.

Compliment set-systems: A368409, connected case of A368094, ranks A367907.

For set-systems we have A368410, connected case of A368095, ranks A367906.

The complement is A368411, connected case of A368097, ranks A355529.

A000110 counts set partitions, non-isomorphic A000041.

A003465 counts covering set-systems, unlabeled A055621.

A007716 counts non-isomorphic multiset partitions, connected A007718.

A058891 counts set-systems, unlabeled A000612, connected A323818.

A283877 counts non-isomorphic set-systems, connected A300913.

Cf. A140637, A302545, A316983, A317533, A319616, A367902, A367905, A368187.

Sequence in context: A071973 A086424 A122121 * A080009 A159336 A240571

Adjacent sequences: A368409 A368410 A368411 * A368413 A368414 A368415

Keyword

nonn,more

Author

Gus Wiseman, Dec 26 2023

Status

approved