cp's OEIS Frontend

A356943 Number of multiset partitions into gapless blocks of a size-n multiset covering an initial interval with weakly decreasing multiplicities.

%I A356943 #7 Sep 10 2022 22:02:02
%S A356943 1,1,4,11,37,101,328,909,2801
%N A356943 Number of multiset partitions into gapless blocks of a size-n multiset covering an initial interval with weakly decreasing multiplicities.
%C A356943 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 A356943 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 A356943 The a(1) = 1 through a(3) = 11 multiset partitions:
%e A356943   {{1}}  {{1,1}}    {{1,1,1}}
%e A356943          {{1,2}}    {{1,1,2}}
%e A356943          {{1},{1}}  {{1,2,3}}
%e A356943          {{1},{2}}  {{1},{1,1}}
%e A356943                     {{1},{1,2}}
%e A356943                     {{1},{2,3}}
%e A356943                     {{2},{1,1}}
%e A356943                     {{3},{1,2}}
%e A356943                     {{1},{1},{1}}
%e A356943                     {{1},{1},{2}}
%e A356943                     {{1},{2},{3}}
%t A356943 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A356943 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A356943 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A356943 nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
%t A356943 Table[Length[Select[Join@@mps/@strnorm[n],And@@nogapQ/@#&]],{n,0,5}]
%Y A356943 A000041 counts integer partitions, strict A000009.
%Y A356943 A000670 counts patterns, ranked by A333217, necklace A019536.
%Y A356943 A011782 counts multisets covering an initial interval.
%Y A356943 Gapless multisets are counted by A034296, ranked by A073491.
%Y A356943 Other conditions: A035310, A063834, A330783, A356934, A356938, A356954.
%Y A356943 Other types: A356233, A356941, A356942, A356944.
%Y A356943 Cf. A055887, A072233, A270995, A304969, A349050, A349055.
%K A356943 nonn,more
%O A356943 0,3
%A A356943 _Gus Wiseman_, Sep 09 2022