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%I #13 Nov 15 2019 21:36:36
%S 1,2,2,1,3,2,1,1,3,1,4,2,1,1,1,2,2,1,3,1,1,3,2,4,1,5,2,1,1,1,1,2,2,1,
%T 1,3,1,1,1,3,1,2,3,2,1,4,1,1,4,2,5,1,6,2,1,1,1,1,1,2,1,2,1,1,2,2,1,1,
%U 1,2,2,2,1,3,1,1,1,1,3,1,1,2,3,1,2,1,3,2,1,1,3,2,2,3,3,1,4,1,1,1,4,1,2,4,2,1,4,3,5,1,1,5,2,6,1,7
%N Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.
%C The *-product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling them together. Every finite positive integer sequence has a unique *-factorization using prime compositions P = {(1), (2), (21), (3), (211), ...}. See A060223 and A228369 for details.
%C These are co-Lyndon compositions, ordered first by sum and then lexicographically. - _Gus Wiseman_, Nov 15 2019
%F Row lengths are A059966(n) = number of prime compositions of n.
%e The prime factorization of (1, 1, 4, 2, 3, 1, 5, 5) is: (11423155) = (1)*(1)*(5)*(5)*(4231). The prime factorizations of the initial terms of A000002 are:
%e (1) = (1)
%e (12) = (1)*(2)
%e (122) = (1)*(2)*(2)
%e (1221) = (1)*(221)
%e (12211) = (1)*(2211)
%e (122112) = (1)*(2)*(2211)
%e (1221121) = (1)*(221121)
%e (12211212) = (1)*(2)*(221121)
%e (122112122) = (1)*(2)*(2)*(221121)
%e (1221121221) = (1)*(221)*(221121)
%e (12211212212) = (1)*(2)*(221)*(221121)
%e (122112122122) = (1)*(2)*(2)*(221)*(221121).
%e Read as a sequence:
%e (1), (2), (21), (3), (211), (31), (4), (2111), (221), (311), (32), (41), (5).
%e Read as a triangle:
%e (1)
%e (2)
%e (21), (3)
%e (211), (31), (4)
%e (2111), (221), (311), (32), (41), (5).
%e Read as a sequence of triangles:
%e 1 2 2 1 2 1 1 2 1 1 1 2 1 1 1 1 2 1 1 1 1 1
%e 3 3 1 2 2 1 2 2 1 1 2 1 2 1 1
%e 4 3 1 1 3 1 1 1 2 2 1 1 1
%e 3 2 3 1 2 2 2 2 1
%e 4 1 3 2 1 3 1 1 1 1
%e 5 4 1 1 3 1 1 2
%e 4 2 3 1 2 1
%e 5 1 3 2 1 1
%e 6 3 2 2
%e 3 3 1
%e 4 1 1 1
%e 4 1 2
%e 4 2 1
%e 4 3
%e 5 1 1
%e 5 2
%e 6 1
%e 7.
%t colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
%t Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n],colynQ],lexsort],{n,5}] (* _Gus Wiseman_, Nov 15 2019 *)
%Y Cf. A000740, A215474, A228369, A277427.
%Y The binary version is A329318.
%Y The binary non-"co" version is A102659.
%Y A sequence listing all Lyndon compositions is A294859.
%Y Numbers whose binary expansion is co-Lyndon are A328596.
%Y Numbers whose binary expansion is co-Lyndon are A275692.
%Y Binary Lyndon words are A001037.
%Y Lyndon compositions are A059966.
%Y Normal Lyndon words are A060223.
%Y Cf. A211097, A211100, A296372, A296373, A298941, A329131, A329312, A329313, A329314, A329324, A329326.
%K nonn,tabf
%O 1,2
%A _Gus Wiseman_, Jan 12 2017
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