cp's OEIS Frontend

A332724 Number of length n - 1 ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.

%I A332724 #11 Apr 17 2021 18:13:56
%S A332724 0,0,1,6,14,32,65,128,243,452,826,1490,2659,4704,8261,14418,25030,
%T A332724 43252,74437,127648,218199,371920,632306,1072486,1815239,3066432,
%U A332724 5170825,8705118,14632958,24562952,41177801,68947520,115313979,192656924,321554986,536191418
%N A332724 Number of length n - 1 ordered set partitions of {1..n} where no element of any block is greater than any element of a non-adjacent consecutive block.
%C A332724 In other words, parts of not-immediately-subsequent blocks are increasing.
%H A332724 Andrew Howroyd, <a href="/A332724/b332724.txt">Table of n, a(n) for n = 0..1000</a>
%H A332724 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A332724 From _Andrew Howroyd_, Apr 17 2021: (Start)
%F A332724 a(n) = A001629(n) + 4*A001629(n+1) + A001629(n+2) for n > 0.
%F A332724 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n > 4.
%F A332724 G.f.: x*(1 + 4*x + x^2)/(1 - x - x^2)^2.
%F A332724 (End)
%e A332724 The a(2) = 1 through a(4) = 14 ordered set partitions:
%e A332724   {{1,2}}  {{1},{2,3}}  {{1},{2},{3,4}}
%e A332724            {{1,2},{3}}  {{1},{2,3},{4}}
%e A332724            {{1,3},{2}}  {{1,2},{3},{4}}
%e A332724            {{2},{1,3}}  {{1},{2,4},{3}}
%e A332724            {{2,3},{1}}  {{1,2},{4},{3}}
%e A332724            {{3},{1,2}}  {{1},{3},{2,4}}
%e A332724                         {{1,3},{2},{4}}
%e A332724                         {{1},{3,4},{2}}
%e A332724                         {{1},{4},{2,3}}
%e A332724                         {{2},{1},{3,4}}
%e A332724                         {{2},{1,3},{4}}
%e A332724                         {{2},{1,4},{3}}
%e A332724                         {{2,3},{1},{4}}
%e A332724                         {{3},{1,2},{4}}
%t A332724 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A332724 Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==n-1&&!MatchQ[#,{___,{___,a_,___},__,{___,b_,___},___}/;a>b]&]],{n,0,8}]
%o A332724 (PARI) \\ here b(n) is A001629(n).
%o A332724 b(n) = {((n+1)*fibonacci(n-1) + (n-1)*fibonacci(n+1))/5}
%o A332724 a(n) = {if(n==0, 0, b(n) + 4*b(n-1) + b(n-2))} \\ _Andrew Howroyd_, Apr 17 2021
%Y A332724 Column k = n - 1 of A332673, which has row-sums A332872.
%Y A332724 Ordered set-partitions are A000670.
%Y A332724 Unimodal compositions are A001523.
%Y A332724 Unimodal normal sequences appear to be A007052.
%Y A332724 Non-unimodal normal sequences are A328509.
%Y A332724 Cf. A000045, A001286, A001629, A056242, A227038, A332577.
%K A332724 nonn
%O A332724 0,4
%A A332724 _Gus Wiseman_, Mar 03 2020
%E A332724 Terms a(9) and beyond from _Andrew Howroyd_, Apr 17 2021