cp's OEIS Frontend

A329141 Number of Lyndon compositions of n that are not weakly increasing.

%I A329141 #4 Nov 10 2019 20:29:11
%S A329141 0,0,0,0,0,1,4,11,28,60,131,263,530,1029,2009,3853,7414,14152,27105,
%T A329141 51755,99069,189558,363468,697302,1340220,2578362,4968001,9582682,
%U A329141 18508226,35784670,69266825,134207336,260290846,505274108,981691926
%N A329141 Number of Lyndon compositions of n that are not weakly increasing.
%C A329141 A Lyndon composition of n is a finite sequence of positive integers summing to n that is lexicographically strictly less than all of its cyclic rotations.
%F A329141 a(n) = A059966(n) - A167934(n).
%e A329141 The a(6) = 1 through a(8) = 11 compositions:
%e A329141   (132)  (142)    (143)
%e A329141          (1132)   (152)
%e A329141          (1213)   (1142)
%e A329141          (11212)  (1214)
%e A329141                   (1232)
%e A329141                   (1322)
%e A329141                   (11132)
%e A329141                   (11213)
%e A329141                   (11312)
%e A329141                   (12122)
%e A329141                   (111212)
%t A329141 neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A329141 aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
%t A329141 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!OrderedQ[#]&&neckQ[#]&&aperQ[#]&]],{n,10}]
%Y A329141 Lyndon compositions are A059966.
%Y A329141 Lyndon compositions that are weakly increasing are A167934.
%Y A329141 Binary Lyndon words are A001037.
%Y A329141 Necklace compositions are A008965.
%Y A329141 Cf. A000031, A000740, A178472, A318731, A328596, A329131, A329145.
%K A329141 nonn
%O A329141 1,7
%A A329141 _Gus Wiseman_, Nov 10 2019