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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

Table of n, a(n) for n=0..17.

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.)