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)
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
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 25 2018
EXTENSIONS
a(11)-a(17) from Gus Wiseman, Jan 18 2019
a(18)-a(21) from Robert Price, Jun 21 2021
STATUS
approved