cp's OEIS Frontend

A332873 Number of non-unimodal, non-co-unimodal sequences of length n covering an initial interval of positive integers.

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