%I #11 Jan 28 2024 18:09:57
%S 0,0,0,0,22,340,3954,44716,536858,7056252,102140970,1622267196,
%T 28090317226,526854073564,10641328363722,230283141084220,
%U 5315654511587498,130370766447282204,3385534661270087178,92801587312544823804,2677687796221222845802,81124824998424994578652
%N Number of non-unimodal, non-co-unimodal sequences of length n covering an initial interval of positive integers.
%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. It is co-unimodal if its negative is unimodal.
%H Andrew Howroyd, <a href="/A332873/b332873.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = A000670(n) + A000225(n) - 2*A007052(n-1) for n > 0. - _Andrew Howroyd_, Jan 28 2024
%e The a(4) = 22 sequences:
%e (1,2,1,2) (2,3,1,3)
%e (1,2,1,3) (2,3,1,4)
%e (1,3,1,2) (2,4,1,3)
%e (1,3,2,3) (3,1,2,1)
%e (1,3,2,4) (3,1,3,2)
%e (1,4,2,3) (3,1,4,2)
%e (2,1,2,1) (3,2,3,1)
%e (2,1,3,1) (3,2,4,1)
%e (2,1,3,2) (3,4,1,2)
%e (2,1,4,3) (4,1,3,2)
%e (2,3,1,2) (4,2,3,1)
%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t Table[Length[Select[Union@@Permutations/@allnorm[n],!unimodQ[#]&&!unimodQ[-#]&]],{n,0,5}]
%o (PARI) seq(n)=Vec( serlaplace(1/(2-exp(x + O(x*x^n)))) - (1 - 6*x + 12*x^2 - 6*x^3)/((1 - x)*(1 - 2*x)*(1 - 4*x + 2*x^2)), -(n+1)) \\ _Andrew Howroyd_, Jan 28 2024
%Y Not requiring non-co-unimodality gives A328509.
%Y Not requiring non-unimodality also gives A328509.
%Y The version for run-lengths of partitions is A332640.
%Y The version for unsorted prime signature is A332643.
%Y The version for compositions is A332870.
%Y Unimodal compositions are A001523.
%Y Unimodal sequences covering an initial interval are A007052.
%Y Non-unimodal permutations are A059204.
%Y Non-unimodal compositions are A115981.
%Y Unimodal compositions covering an initial interval are A227038.
%Y Numbers whose unsorted prime signature is not unimodal are A332282.
%Y Numbers whose negated prime signature is not unimodal are A332642.
%Y Compositions whose run-lengths are not unimodal are A332727.
%Y Non-unimodal compositions covering an initial interval are A332743.
%Y Cf. A000225, A000670, A060223, A072704, A329398, A332281, A332284, A332577, A332578, A332639, A332672, A332834.
%K nonn
%O 0,5
%A _Gus Wiseman_, Mar 03 2020
%E a(9) onwards from _Andrew Howroyd_, Jan 28 2024
|