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%I #12 Jun 20 2021 11:07:30
%S 0,0,0,0,0,1,2,7,16,37,76,166,328,669,1326,2626,5138,10104,19680,
%T 38442,74822,145715,283424,551721,1073224
%N Number of Lyndon compositions of n whose reverse is not a co-Lyndon composition.
%C A Lyndon composition of n is a finite sequence summing to n that is lexicographically strictly less than all of its cyclic rotations. A co-Lyndon composition of n is a finite sequence summing to n that is lexicographically strictly greater than all of its cyclic rotations.
%e The a(6) = 1 through a(9) = 16 compositions:
%e (132) (142) (143) (153)
%e (1132) (152) (162)
%e (1142) (243)
%e (1232) (1143)
%e (1322) (1152)
%e (11132) (1242)
%e (11312) (1332)
%e (1422)
%e (11142)
%e (11232)
%e (11322)
%e (11412)
%e (12132)
%e (111132)
%e (111312)
%e (112212)
%t lynQ[q_]:=Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
%t colynQ[q_]:=Array[Union[{RotateRight[q,#1],q}]=={RotateRight[q,#1],q}&,Length[q]-1,1,And];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],lynQ[#]&&!colynQ[Reverse[#]]&]],{n,15}]
%Y Lyndon and co-Lyndon compositions are counted by A059966.
%Y Numbers whose reversed binary expansion is Lyndon are A328596.
%Y Numbers whose binary expansion is co-Lyndon are A275692.
%Y Lyndon compositions that are not weakly increasing are A329141.
%Y Cf. A000740, A001037, A008965, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326.
%K nonn,more
%O 1,7
%A _Gus Wiseman_, Nov 11 2019
%E a(21)-a(25) from _Robert Price_, Jun 20 2021
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