cp's OEIS Frontend

A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.

%I A319646 #13 Jun 21 2021 23:30:37
%S A319646 1,1,1,4,4,9,17,28,41,75,122,192,314,484,771,1216,1861,2848,4395,6610,
%T A319646 10037
%N A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets.
%C A319646 The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
%C A319646 The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%C A319646 From _Gus Wiseman_, Jan 17 2019: (Start)
%C A319646 Also the number of plane partitions of n with no repeated rows or columns. For example, the a(6) = 17 plane partitions are:
%C A319646   6   51   42   321
%C A319646 .
%C A319646   5   4   41   31   32   31   22   221   211
%C A319646   1   2   1    2    1    11   2    1     11
%C A319646 .
%C A319646   3   21   21   111
%C A319646   2   2    11   11
%C A319646   1   1    1    1
%C A319646 (End)
%e A319646 Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 chains:
%e A319646 1: {{1}}
%e A319646 2: {{1,1}}
%e A319646 3: {{1,1,1}}
%e A319646    {{1,2,2}}
%e A319646    {{1},{1,1}}
%e A319646    {{2},{1,2}}
%e A319646 4: {{1,1,1,1}}
%e A319646    {{1,2,2,2}}
%e A319646    {{1},{1,1,1}}
%e A319646    {{2},{1,2,2}}
%e A319646 5: {{1,1,1,1,1}}
%e A319646    {{1,1,2,2,2}}
%e A319646    {{1,2,2,2,2}}
%e A319646    {{1},{1,1,1,1}}
%e A319646    {{2},{1,1,2,2}}
%e A319646    {{2},{1,2,2,2}}
%e A319646    {{1,1},{1,1,1}}
%e A319646    {{1,2},{1,2,2}}
%e A319646    {{2,2},{1,2,2}}
%t A319646 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A319646 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A319646 ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
%t A319646 Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[UnsameQ@@#,UnsameQ@@Transpose[PadRight[#]],And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,IntegerPartitions[n]}],{n,10}] (* _Gus Wiseman_, Jan 18 2019 *)
%Y A319646 Cf. A000219, A003293, A007716, A059201, A283877, A316980, A316983, A318099, A319558, A319616-A319646.
%Y A319646 Cf. A000085, A138178, A323436.
%K A319646 nonn,more
%O A319646 0,4
%A A319646 _Gus Wiseman_, Sep 25 2018
%E A319646 a(11)-a(17) from _Gus Wiseman_, Jan 18 2019
%E A319646 a(18)-a(21) from _Robert Price_, Jun 21 2021