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A332669 Number of compositions of n whose negation is not unimodal.

%I A332669 #16 Feb 16 2025 08:33:59
%S A332669 0,0,0,0,1,3,11,28,71,165,372,807,1725,3611,7481,15345,31274,63392,
%T A332669 128040,257865,518318,1040277,2085714,4178596,8367205,16748151,
%U A332669 33515214,67056139,134147231,268341515,536746350,1073577185,2147266984,4294683056,8589563136,17179385180
%N A332669 Number of compositions of n whose negation is not unimodal.
%C A332669 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332669 A composition of n is a finite sequence of positive integers summing to n.
%H A332669 Andrew Howroyd, <a href="/A332669/b332669.txt">Table of n, a(n) for n = 0..1000</a>
%H A332669 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F A332669 a(n) + A332578(n) = 2^(n - 1) for n > 0.
%e A332669 The a(4) = 1 through a(6) = 11 compositions:
%e A332669   (121)  (131)   (132)
%e A332669          (1121)  (141)
%e A332669          (1211)  (231)
%e A332669                  (1131)
%e A332669                  (1212)
%e A332669                  (1221)
%e A332669                  (1311)
%e A332669                  (2121)
%e A332669                  (11121)
%e A332669                  (11211)
%e A332669                  (12111)
%t A332669 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332669 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[-#]&]],{n,0,10}]
%Y A332669 The strict case is A072707.
%Y A332669 The complement is counted by A332578.
%Y A332669 The version for run-lengths of partitions is A332639.
%Y A332669 The version for unsorted prime signature is A332642.
%Y A332669 The version for 0-appended first-differences of partitions is A332744.
%Y A332669 The case that is not unimodal either is A332870.
%Y A332669 Unimodal compositions are A001523.
%Y A332669 Non-unimodal permutations are A059204.
%Y A332669 Non-unimodal compositions are A115981.
%Y A332669 Non-unimodal normal sequences are A328509.
%Y A332669 Numbers whose unsorted prime signature is not unimodal are A332282.
%Y A332669 A triangle for compositions with unimodal negation is A332670.
%Y A332669 Cf. A007052, A072706, A227038, A329398, A332281, A332284, A332638, A332728, A332742, A332832.
%K A332669 nonn
%O A332669 0,6
%A A332669 _Gus Wiseman_, Feb 28 2020
%E A332669 Terms a(21) and beyond from _Andrew Howroyd_, Mar 01 2020