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 A319646 Number of non-isomorphic weight-n chains of distinct multisets whose dual is also a chain of distinct multisets. 37
 1, 1, 1, 4, 4, 9, 17, 28, 41, 75, 122, 192, 314, 484, 771, 1216, 1861, 2848 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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}}. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices. From Gus Wiseman, Jan 17 2019: (Start) Also the number of plane partitions of n with no repeated rows or columns. For example, the a(6) = 17 plane partitions are:   6   51   42   321 .   5   4   41   31   32   31   22   221   211   1   2   1    2    1    11   2    1     11 .   3   21   21   111   2   2    11   11   1   1    1    1 (End) LINKS EXAMPLE Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 chains: 1: {{1}} 2: {{1,1}} 3: {{1,1,1}}    {{1,2,2}}    {{1},{1,1}}    {{2},{1,2}} 4: {{1,1,1,1}}    {{1,2,2,2}}    {{1},{1,1,1}}    {{2},{1,2,2}} 5: {{1,1,1,1,1}}    {{1,1,2,2,2}}    {{1,2,2,2,2}}    {{1},{1,1,1,1}}    {{2},{1,1,2,2}}    {{2},{1,2,2,2}}    {{1,1},{1,1,1}}    {{1,2},{1,2,2}}    {{2,2},{1,2,2}} MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]]; ptnplane[n_]:=Union[Map[Reverse@*primeMS, Join@@Permutations/@facs[n], {2}]]; 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 *) CROSSREFS Cf. A000219, A003293, A007716, A059201, A283877, A316980, A316983, A318099, A319558, A319616-A319646. Cf. A000085, A138178, A323436. Sequence in context: A202670 A203003 A339427 * A214826 A135065 A067553 Adjacent sequences:  A319643 A319644 A319645 * A319647 A319648 A319649 KEYWORD nonn,more AUTHOR Gus Wiseman, Sep 25 2018 EXTENSIONS a(11)-a(17) from Gus Wiseman, Jan 18 2019 STATUS approved

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Last modified April 21 13:37 EDT 2021. Contains 343153 sequences. (Running on oeis4.)