A332874 - OEIS

%I #13 Apr 17 2021 03:42:05

%S 0,0,0,0,0,0,0,0,0,0,10,10,20,30,50,150,180,290,420,630,860,1828,2168,

%T 3326,4514,6530,8576,12188,20096,25314,35576,48062,65592,86752,117222,

%U 152060,237590,292346,402798,524596,711270,910606,1221204,1554382,2044460,2927124

%N Number of strict compositions of n that are neither unimodal nor is their negation.

%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

%C A composition of n is a finite sequence of positive integers summing to n. It is strict if there are not repeated parts.

%H Andrew Howroyd, <a href="/A332874/b332874.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>

%F G.f.: Sum_{k>=4} (k! - 2^k + 2) * [y^k](Product_{j>=1} 1 + y*x^j). - _Andrew Howroyd_, Apr 16 2021

%e The a(10) = 10 through a(12) = 20 compositions:

%e   (1,3,2,4)  (1,3,2,5)  (1,3,2,6)

%e   (1,4,2,3)  (1,5,2,3)  (1,4,2,5)

%e   (2,1,4,3)  (2,1,5,3)  (1,5,2,4)

%e   (2,3,1,4)  (2,3,1,5)  (1,6,2,3)

%e   (2,4,1,3)  (2,5,1,3)  (2,1,5,4)

%e   (3,1,4,2)  (3,1,5,2)  (2,1,6,3)

%e   (3,2,4,1)  (3,2,5,1)  (2,3,1,6)

%e   (3,4,1,2)  (3,5,1,2)  (2,4,1,5)

%e   (4,1,3,2)  (5,1,3,2)  (2,5,1,4)

%e   (4,2,3,1)  (5,2,3,1)  (2,6,1,3)

%e                         (3,1,6,2)

%e                         (3,2,6,1)

%e                         (3,6,1,2)

%e                         (4,1,5,2)

%e                         (4,2,5,1)

%e                         (4,5,1,2)

%e                         (5,1,4,2)

%e                         (5,2,4,1)

%e                         (6,1,3,2)

%e                         (6,2,3,1)

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!unimodQ[#]&&!unimodQ[-#]&]],{n,0,20}]

%o (PARI) seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=4, n, (k! - 2^k + 2)*polcoef(p,k,y)), -(n+1))} \\ _Andrew Howroyd_, Apr 16 2021

%Y The non-strict version for unsorted prime signature is A332643.

%Y The non-strict version is A332870.

%Y Unimodal compositions are A001523.

%Y Non-unimodal compositions are A115981.

%Y Non-unimodal normal sequences are A328509.

%Y Compositions whose negation is unimodal are A332578.

%Y Compositions whose negation is not unimodal are A332669.

%Y Compositions with neither weakly increasing nor weakly decreasing run-lengths are A332833.

%Y Compositions with weakly increasing or weakly decreasing run-lengths are A332835.

%Y Cf. A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332640, A332641, A332745, A332746, A332831, A332834.

%K nonn

%O 0,11

%A _Gus Wiseman_, Mar 04 2020

%E Terms a(21) and beyond from _Andrew Howroyd_, Apr 16 2021