|
%I #22 Jan 28 2024 18:10:02
%S 0,0,0,3,41,425,4287,45941,541219,7071501,102193755,1622448861,
%T 28090940363,526856206877,10641335658891,230283166014653,
%U 5315654596751659,130370766738143517,3385534662263335179,92801587315936355325,2677687796232803000171,81124824998464533181661
%N Number of non-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.
%H Andrew Howroyd, <a href="/A328509/b328509.txt">Table of n, a(n) for n = 0..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F a(n) = A000670(n) - A007052(n-1) for n > 0. - _Andrew Howroyd_, Jan 28 2024
%e The a(3) = 3 sequences are (2,1,2), (2,1,3), (3,1,2).
%e The a(4) = 41 sequences:
%e (1212) (2113) (2134) (2413) (3142) (3412)
%e (1213) (2121) (2143) (3112) (3212) (4123)
%e (1312) (2122) (2212) (3121) (3213) (4132)
%e (1323) (2123) (2213) (3122) (3214) (4213)
%e (1324) (2131) (2312) (3123) (3231) (4231)
%e (1423) (2132) (2313) (3124) (3241) (4312)
%e (2112) (2133) (2314) (3132) (3312)
%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[#]&]],{n,0,5}]
%o (PARI) seq(n)=Vec( serlaplace(1/(2-exp(x + O(x*x^n)))) - (1 - 3*x + x^2)/(1 - 4*x + 2*x^2), -(n+1)) \\ _Andrew Howroyd_, Jan 28 2024
%Y Not requiring non-unimodality gives A000670.
%Y The complement is counted by A007052.
%Y The case where the negation is not unimodal either is A332873.
%Y Unimodal compositions are A001523.
%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 Covering partitions with unimodal run-lengths are A332577.
%Y Non-unimodal compositions covering an initial interval are A332743.
%Y Cf. A060223, A255906, A332281, A332284, A332639, A332672, A332834, A332870.
%K nonn
%O 0,4
%A _Gus Wiseman_, Feb 19 2020
%E a(9) from _Robert Price_, Jun 19 2021
%E a(10) onwards from _Andrew Howroyd_, Jan 28 2024
| 