|
%I #18 Jun 29 2021 06:38:56
%S 0,0,0,0,0,1,5,17,49,126,303,694,1536,3312,7009,14619,30164,61732,
%T 125568,254246,513048,1032696,2074875,4163256,8345605,16717996,
%U 33473334,66998380,134067959,268233386,536599508,1073378850,2147000209
%N The number of compositions of n which cannot be viewed as stacks.
%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. A composition of n is a finite sequence of positive integers summing to n. - _Gus Wiseman_, Mar 05 2020
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F a(n) = A011782(n) - A001523(n).
%e a(5) = 1 counting (212}.
%e a(6) = 5 counting {1212, 2112,2121,213,312}.
%e a(7) = 17 counting {11212, 12112,12121, 21211, 21121, 21112, 2122, 2212, 2113, 3112, 2131, 3121, 1213, 1312, 412, 214, 313}.
%e a(8) = 49 = 128 - 79.
%e a(9) = 126 = 256 - 130.
%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],!unimodQ[#]&]],{n,0,10}] (* _Gus Wiseman_, Mar 05 2020 *)
%Y Cf. A011782, A115982.
%Y The complement is counted by A001523.
%Y The strict case is A072707.
%Y The case covering an initial interval is A332743.
%Y The version whose negation is not unimodal either is A332870.
%Y Non-unimodal permutations are A059204.
%Y Non-unimodal normal sequences are A328509.
%Y Partitions with non-unimodal run-lengths are A332281.
%Y Numbers whose prime signature is not unimodal are A332282.
%Y Partitions whose 0-appended first differences are not unimodal are A332284.
%Y Non-unimodal permutations of the prime indices of n are A332671.
%Y Cf. A007052, A072704, A227038, A329398, A332280, A332283, A332672, A332578, A332669, A332834.
%K easy,nonn
%O 0,7
%A _Alford Arnold_, Feb 12 2006
%E More terms from Brian Kuehn (brk158(AT)psu.edu), Apr 20 2006
%E a(25) corrected by _Georg Fischer_, Jun 29 2021
|
