A356934 - OEIS

A356934

Number of multisets of odd-size multisets whose multiset union is a size-n multiset covering an initial interval with weakly decreasing multiplicities.

%I #6 Sep 10 2022 14:02:27

%S 1,1,2,6,17,46,166,553,2093

%N Number of multisets of odd-size multisets whose multiset union is a size-n multiset covering an initial interval with 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) = 17 multiset partitions:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%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 Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.

%Y Other conditions: A035310, A063834, A330783, A356938, A356943, A356954.

%Y Other types: A050330, A356932, A356933, A356935.

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

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Sep 09 2022