cp's OEIS Frontend

A356954 Number of multisets of multisets, each covering an initial interval, whose multiset union is of size n and has weakly decreasing multiplicities.

%I A356954 #7 Sep 13 2022 13:07:01
%S A356954 1,1,3,6,15,30,71,145,325,680
%N A356954 Number of multisets of multisets, each covering an initial interval, whose multiset union is of size n and has weakly decreasing multiplicities.
%H A356954 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 A356954 The a(1) = 1 through a(4) = 15 multiset partitions:
%e A356954   {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
%e A356954          {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
%e A356954          {{1},{1}}  {{1,2,3}}      {{1,1,2,2}}
%e A356954                     {{1},{1,1}}    {{1,1,2,3}}
%e A356954                     {{1},{1,2}}    {{1,2,3,4}}
%e A356954                     {{1},{1},{1}}  {{1},{1,1,1}}
%e A356954                                    {{1,1},{1,1}}
%e A356954                                    {{1},{1,1,2}}
%e A356954                                    {{1,1},{1,2}}
%e A356954                                    {{1},{1,2,2}}
%e A356954                                    {{1},{1,2,3}}
%e A356954                                    {{1,2},{1,2}}
%e A356954                                    {{1},{1},{1,1}}
%e A356954                                    {{1},{1},{1,2}}
%e A356954                                    {{1},{1},{1},{1}}
%t A356954 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A356954 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A356954 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A356954 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A356954 Table[Length[Select[Join@@mps/@strnorm[n],And@@normQ/@#&]],{n,0,5}]
%Y A356954 For unrestricted multiplicities we have A034691.
%Y A356954 A000041 counts integer partitions, strict A000009.
%Y A356954 A000670 counts patterns, ranked by A333217, necklace A019536.
%Y A356954 A011782 counts multisets covering an initial interval.
%Y A356954 Other conditions: A035310, A063834, A330783, A356934, A356938, A356943.
%Y A356954 Other types: A055932, A089259, A356945, A356955.
%Y A356954 Cf. A055887, A072233, A270995, A304969, A349050, A349055, A356942.
%K A356954 nonn,more
%O A356954 0,3
%A A356954 _Gus Wiseman_, Sep 09 2022