cp's OEIS Frontend

A328509 Number of non-unimodal sequences of length n covering an initial interval of positive integers.

%I A328509 #23 Feb 16 2025 08:33:58
%S A328509 0,0,0,3,41,425,4287,45941,541219,7071501,102193755,1622448861,
%T A328509 28090940363,526856206877,10641335658891,230283166014653,
%U A328509 5315654596751659,130370766738143517,3385534662263335179,92801587315936355325,2677687796232803000171,81124824998464533181661
%N A328509 Number of non-unimodal sequences of length n covering an initial interval of positive integers.
%C A328509 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A328509 Andrew Howroyd, <a href="/A328509/b328509.txt">Table of n, a(n) for n = 0..200</a>
%H A328509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F A328509 a(n) = A000670(n) - A007052(n-1) for n > 0. - _Andrew Howroyd_, Jan 28 2024
%e A328509 The a(3) = 3 sequences are (2,1,2), (2,1,3), (3,1,2).
%e A328509 The a(4) = 41 sequences:
%e A328509   (1212)  (2113)  (2134)  (2413)  (3142)  (3412)
%e A328509   (1213)  (2121)  (2143)  (3112)  (3212)  (4123)
%e A328509   (1312)  (2122)  (2212)  (3121)  (3213)  (4132)
%e A328509   (1323)  (2123)  (2213)  (3122)  (3214)  (4213)
%e A328509   (1324)  (2131)  (2312)  (3123)  (3231)  (4231)
%e A328509   (1423)  (2132)  (2313)  (3124)  (3241)  (4312)
%e A328509   (2112)  (2133)  (2314)  (3132)  (3312)
%t A328509 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A328509 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A328509 Table[Length[Select[Union@@Permutations/@allnorm[n],!unimodQ[#]&]],{n,0,5}]
%o A328509 (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 A328509 Not requiring non-unimodality gives A000670.
%Y A328509 The complement is counted by A007052.
%Y A328509 The case where the negation is not unimodal either is A332873.
%Y A328509 Unimodal compositions are A001523.
%Y A328509 Non-unimodal permutations are A059204.
%Y A328509 Non-unimodal compositions are A115981.
%Y A328509 Unimodal compositions covering an initial interval are A227038.
%Y A328509 Numbers whose unsorted prime signature is not unimodal are A332282.
%Y A328509 Covering partitions with unimodal run-lengths are A332577.
%Y A328509 Non-unimodal compositions covering an initial interval are A332743.
%Y A328509 Cf. A060223, A255906, A332281, A332284, A332639, A332672, A332834, A332870.
%K A328509 nonn
%O A328509 0,4
%A A328509 _Gus Wiseman_, Feb 19 2020
%E A328509 a(9) from _Robert Price_, Jun 19 2021
%E A328509 a(10) onwards from _Andrew Howroyd_, Jan 28 2024