A356954 - OEIS

A356954

Number of multisets of multisets, each covering an initial interval, whose multiset union is of size n and has weakly decreasing multiplicities.

%I #7 Sep 13 2022 13:07:01

%S 1,1,3,6,15,30,71,145,325,680

%N Number of multisets of multisets, each covering an initial interval, whose multiset union is of size n and has weakly decreasing multiplicities.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vR-C_picqWlu0KOguRGWaPjhS2HY7m43aGXGDcolDh4Qtyy-pu2lkq5mbHAbiMSyQoiIESG2mCGtc2j/pub">Counting and ranking classes of multiset partitions related to gapless multisets</a>

%e The a(1) = 1 through a(4) = 15 multiset partitions:

%e {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}

%e {{1,2}} {{1,1,2}} {{1,1,1,2}}

%e {{1},{1}} {{1,2,3}} {{1,1,2,2}}

%e {{1},{1,1}} {{1,1,2,3}}

%e {{1},{1,2}} {{1,2,3,4}}

%e {{1},{1},{1}} {{1},{1,1,1}}

%e {{1,1},{1,1}}

%e {{1},{1,1,2}}

%e {{1,1},{1,2}}

%e {{1},{1,2,2}}

%e {{1},{1,2,3}}

%e {{1,2},{1,2}}

%e {{1},{1},{1,1}}

%e {{1},{1},{1,2}}

%e {{1},{1},{1},{1}}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];

%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];

%t Table[Length[Select[Join@@mps/@strnorm[n],And@@normQ/@#&]],{n,0,5}]

%Y For unrestricted multiplicities we have A034691.

%Y A000041 counts integer partitions, strict A000009.

%Y A000670 counts patterns, ranked by A333217, necklace A019536.

%Y A011782 counts multisets covering an initial interval.

%Y Other conditions: A035310, A063834, A330783, A356934, A356938, A356943.

%Y Other types: A055932, A089259, A356945, A356955.

%Y Cf. A055887, A072233, A270995, A304969, A349050, A349055, A356942.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Sep 09 2022