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%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
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