A332872 - OEIS

%I #5 Mar 06 2020 22:39:56

%S 1,1,3,10,34,116,396,1352,4616,15760

%N Number of ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.

%C After initial terms, first differs from A291292 at a(7) = 1352, A291292(8) = 1353.

%C Conjectured to be the same as A007052, shifted right once.

%e The a(1) = 1 through a(3) = 10 ordered set partitions:

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

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

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

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

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

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

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

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

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

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

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

%t Table[Length[Select[Join@@Permutations/@sps[Range[n]],!MatchQ[#,{___,{___,a_,___},__,{___,b_,___},___}/;a>b]&]],{n,0,5}]

%Y Row sums of A332673.

%Y Set partitions are A000110.

%Y Ordered set-partitions are A000670.

%Y Unimodal sequences covering an initial interval are A007052.

%Y Cf. A001523, A056242, A097805, A328509, A332280, A332283, A332294, A332724.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Mar 06 2020