A332724 - OEIS

%I #11 Apr 17 2021 18:13:56

%S 0,0,1,6,14,32,65,128,243,452,826,1490,2659,4704,8261,14418,25030,

%T 43252,74437,127648,218199,371920,632306,1072486,1815239,3066432,

%U 5170825,8705118,14632958,24562952,41177801,68947520,115313979,192656924,321554986,536191418

%N 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 In other words, parts of not-immediately-subsequent blocks are increasing.

%H Andrew Howroyd, <a href="/A332724/b332724.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).

%F From _Andrew Howroyd_, Apr 17 2021: (Start)

%F a(n) = A001629(n) + 4*A001629(n+1) + A001629(n+2) for n > 0.

%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n > 4.

%F G.f.: x*(1 + 4*x + x^2)/(1 - x - x^2)^2.

%F (End)

%e The a(2) = 1 through a(4) = 14 ordered set partitions:

%e   {{1,2}}  {{1},{2,3}}  {{1},{2},{3,4}}

%e            {{1,2},{3}}  {{1},{2,3},{4}}

%e            {{1,3},{2}}  {{1,2},{3},{4}}

%e            {{2},{1,3}}  {{1},{2,4},{3}}

%e            {{2,3},{1}}  {{1,2},{4},{3}}

%e            {{3},{1,2}}  {{1},{3},{2,4}}

%e                         {{1,3},{2},{4}}

%e                         {{1},{3,4},{2}}

%e                         {{1},{4},{2,3}}

%e                         {{2},{1},{3,4}}

%e                         {{2},{1,3},{4}}

%e                         {{2},{1,4},{3}}

%e                         {{2,3},{1},{4}}

%e                         {{3},{1,2},{4}}

%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]],Length[#]==n-1&&!MatchQ[#,{___,{___,a_,___},__,{___,b_,___},___}/;a>b]&]],{n,0,8}]

%o (PARI) \\ here b(n) is A001629(n).

%o b(n) = {((n+1)*fibonacci(n-1) + (n-1)*fibonacci(n+1))/5}

%o a(n) = {if(n==0, 0, b(n) + 4*b(n-1) + b(n-2))} \\ _Andrew Howroyd_, Apr 17 2021

%Y Column k = n - 1 of A332673, which has row-sums A332872.

%Y Ordered set-partitions are A000670.

%Y Unimodal compositions are A001523.

%Y Unimodal normal sequences appear to be A007052.

%Y Non-unimodal normal sequences are A328509.

%Y Cf. A000045, A001286, A001629, A056242, A227038, A332577.

%K nonn

%O 0,4

%A _Gus Wiseman_, Mar 03 2020

%E Terms a(9) and beyond from _Andrew Howroyd_, Apr 17 2021