%I #7 Mar 12 2024 16:33:25
%S 1,1,2,4,10,23,62,165,475,1400,4334
%N Number of non-isomorphic multiset partitions of weight n such that only one set can be obtained by choosing a different element of each block.
%C A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements.
%e The multiset partition {{3},{1,3},{2,3}} has unique choice (3,1,2) so is counted under a(5).
%e Representatives of the a(1) = 1 through a(5) = 23 multiset partitions:
%e {1} {11} {111} {1111} {11111}
%e {1}{2} {1}{22} {1}{122} {11}{122}
%e {2}{12} {11}{22} {1}{1222}
%e {1}{2}{3} {12}{12} {11}{222}
%e {1}{222} {12}{122}
%e {12}{22} {1}{2222}
%e {2}{122} {12}{222}
%e {1}{2}{33} {2}{1122}
%e {1}{3}{23} {2}{1222}
%e {1}{2}{3}{4} {22}{122}
%e {1}{2}{233}
%e {1}{22}{33}
%e {1}{23}{23}
%e {1}{2}{333}
%e {1}{23}{33}
%e {1}{3}{233}
%e {2}{12}{33}
%e {2}{13}{23}
%e {2}{3}{123}
%e {3}{13}{23}
%e {1}{2}{3}{44}
%e {1}{2}{4}{34}
%e {1}{2}{3}{4}{5}
%Y For existence we have A368098, complement A368097.
%Y Multisets of this type are ranked by A368101, see also A368100, A355529.
%Y Subsets of this type are counted by A370584, see also A370582, A370583.
%Y Maximal sets of this type are counted by A370585.
%Y Partitions of this type are counted by A370594, see also A370592, A370593.
%Y Subsets of this type are also counted by A370638, see also A370636, A370637.
%Y Factorizations of this type are A370645, see also A368414, A368413.
%Y Set-systems of this type are A370818, see also A367902, A367903.
%Y A000110 counts set partitions, non-isomorphic A000041.
%Y A001055 counts factorizations, strict A045778.
%Y A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y Cf. A000612, A055621, A283877, A300913, A302545, A316983, A319616, A330223, A368095, A368412, A368422.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Mar 12 2024