A329324 - OEIS

%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