login
Number of non-isomorphic series/singleton-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n atoms.
8

%I #6 Dec 27 2019 08:58:01

%S 1,1,1,3,17,100,755

%N Number of non-isomorphic series/singleton-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n atoms.

%C A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).

%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 17 trees:

%e {1} {1,2} {1,2,3} {1,2,3,4}

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

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

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

%e {{1},{2,3,4}}

%e {{1,2},{3,4}}

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

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

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

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

%e {{1},{2},{3,4}}

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

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

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

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

%e {{1},{{2},{3,4}}}

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

%Y The non-singleton-reduced version is A330624.

%Y The generalization where leaves are multisets is A330470.

%Y A labeled version is A330628 (strongly normal).

%Y The case with all atoms distinct is A004114.

%Y The balanced version is A330668.

%Y Cf. A000669, A004111, A005804, A007716, A141268, A330465, A330625, A330627, A330654, A330663, A330677.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Dec 26 2019