cp's OEIS Frontend

A356938 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers with weakly decreasing multiplicities.

%I A356938 #5 Sep 10 2022 21:05:38
%S A356938 1,1,3,7,18,41,101,228,538,1209
%N A356938 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers with weakly decreasing multiplicities.
%C A356938 An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.
%H A356938 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 A356938 The a(1) = 1 through a(4) = 18 multiset partitions:
%e A356938   {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
%e A356938          {{1},{1}}  {{1},{1,2}}    {{1},{1,2,3}}
%e A356938          {{1},{2}}  {{1},{2,3}}    {{1,2},{1,2}}
%e A356938                     {{3},{1,2}}    {{1},{2,3,4}}
%e A356938                     {{1},{1},{1}}  {{1,2},{3,4}}
%e A356938                     {{1},{1},{2}}  {{4},{1,2,3}}
%e A356938                     {{1},{2},{3}}  {{1},{1},{1,2}}
%e A356938                                    {{1},{1},{2,3}}
%e A356938                                    {{1},{2},{1,2}}
%e A356938                                    {{1},{2},{3,4}}
%e A356938                                    {{1},{3},{1,2}}
%e A356938                                    {{1},{4},{2,3}}
%e A356938                                    {{3},{4},{1,2}}
%e A356938                                    {{1},{1},{1},{1}}
%e A356938                                    {{1},{1},{1},{2}}
%e A356938                                    {{1},{1},{2},{2}}
%e A356938                                    {{1},{1},{2},{3}}
%e A356938                                    {{1},{2},{3},{4}}
%t A356938 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A356938 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A356938 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A356938 chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
%t A356938 Table[Length[Select[Join@@mps/@strnorm[n],And@@chQ/@#&]],{n,0,5}]
%Y A356938 A000041 counts integer partitions, strict A000009.
%Y A356938 A000670 counts patterns, ranked by A333217, necklace A019536.
%Y A356938 A011782 counts multisets covering an initial interval.
%Y A356938 Intervals are counted by A000012, A001227, ranked by A073485.
%Y A356938 Other conditions: A035310, A063834, A330783, A356934.
%Y A356938 Other types: A107742, A356936, A356937, A356939, A356943, A356954.
%Y A356938 Cf. A055887, A072233, A270995, A304969, A349050, A349055.
%K A356938 nonn,more
%O A356938 0,3
%A A356938 _Gus Wiseman_, Sep 09 2022