A356943 - OEIS

%I #7 Sep 10 2022 22:02:02

%S 1,1,4,11,37,101,328,909,2801

%N Number of multiset partitions into gapless blocks of a size-n multiset covering an initial interval with weakly decreasing multiplicities.

%C A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.

%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(3) = 11 multiset partitions:

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

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

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

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

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

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

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

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

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

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

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

%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 nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];

%t Table[Length[Select[Join@@mps/@strnorm[n],And@@nogapQ/@#&]],{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 Gapless multisets are counted by A034296, ranked by A073491.

%Y Other conditions: A035310, A063834, A330783, A356934, A356938, A356954.

%Y Other types: A356233, A356941, A356942, A356944.

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

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Sep 09 2022