A356957 - OEIS

%I #7 Sep 14 2022 08:26:10

%S 1,1,1,3,2,4,7,7,8,13,20,19,27,30,42,60,63,75,99,112,141,191,205,248,

%T 296,357,408,513,617,696,831,969,1117,1337,1523,1797,2171,2420,2805,

%U 3265,3772,4289,5013,5661,6579,7679,8615,9807,11335,12799,14581

%N Number of set partitions of strict integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1.

%e The a(1) = 1 through a(6) = 7 set partitions:

%e   {{1}}  {{2}}  {{3}}      {{4}}      {{5}}      {{6}}

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

%e                 {{1},{2}}             {{1},{4}}  {{1},{5}}

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

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

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

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

%t chQ[y_] := Length[y] <= 1 || Union[Differences[y]] == {1};

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Length[Select[Join@@sps/@Reverse/@Select[IntegerPartitions[n], UnsameQ@@#&],And@@chQ/@#&]],{n,0,15}]

%Y Intervals are counted by A000012, A001227, ranked by A073485.

%Y The initial version is A010054.

%Y For set partitions of {1..n} we have A011782.

%Y The non-strict version is A107742

%Y Not restricting to intervals gives A294617.

%Y More types: A356936, A356937, A356938, A356939, A356940.

%Y A000041 counts integer partitions, strict A000009.

%Y A000110 counts set partitions.

%Y A001970 counts multiset partitions of integer partitions.

%Y A356941 counts multiset partitions of integer partitions w/ gapless blocks.

%Y Cf. A001055, A055887, A061260, A270995, A304969, A349050, A349055.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 13 2022