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A329359 Irregular triangle read by rows where row n gives the lengths of the factors in the co-Lyndon factorization of the binary expansion of n.

%I A329359 #14 Feb 20 2020 13:55:19
%S A329359 1,2,1,1,3,2,1,3,1,1,1,4,3,1,2,2,2,1,1,4,3,1,4,1,1,1,1,5,4,1,3,2,3,1,
%T A329359 1,5,2,2,1,2,3,2,1,1,1,5,4,1,5,3,1,1,5,4,1,5,1,1,1,1,1,6,5,1,4,2,4,1,
%U A329359 1,3,3,3,2,1,3,3,3,1,1,1,6,5,1,2,2,2,2
%N A329359 Irregular triangle read by rows where row n gives the lengths of the factors in the co-Lyndon factorization of the binary expansion of n.
%C A329359 The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).
%e A329359 Triangle begins:
%e A329359    1: (1)       21: (221)      41: (51)       61: (51)
%e A329359    2: (2)       22: (23)       42: (222)      62: (6)
%e A329359    3: (11)      23: (2111)     43: (2211)     63: (111111)
%e A329359    4: (3)       24: (5)        44: (24)       64: (7)
%e A329359    5: (21)      25: (41)       45: (231)      65: (61)
%e A329359    6: (3)       26: (5)        46: (24)       66: (52)
%e A329359    7: (111)     27: (311)      47: (21111)    67: (511)
%e A329359    8: (4)       28: (5)        48: (6)        68: (43)
%e A329359    9: (31)      29: (41)       49: (51)       69: (421)
%e A329359   10: (22)      30: (5)        50: (6)        70: (43)
%e A329359   11: (211)     31: (11111)    51: (411)      71: (4111)
%e A329359   12: (4)       32: (6)        52: (6)        72: (7)
%e A329359   13: (31)      33: (51)       53: (51)       73: (331)
%e A329359   14: (4)       34: (42)       54: (33)       74: (322)
%e A329359   15: (1111)    35: (411)      55: (3111)     75: (3211)
%e A329359   16: (5)       36: (33)       56: (6)        76: (34)
%e A329359   17: (41)      37: (321)      57: (51)       77: (331)
%e A329359   18: (32)      38: (33)       58: (6)        78: (34)
%e A329359   19: (311)     39: (3111)     59: (411)      79: (31111)
%e A329359   20: (5)       40: (6)        60: (6)        80: (7)
%e A329359 For example, 45 has binary expansion (101101), with co-Lyndon factorization (10)(110)(1), so row n = 45 is (2,3,1).
%t A329359 colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t A329359 colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
%t A329359 Table[Length/@colynfac[If[n==0,{},IntegerDigits[n,2]]],{n,30}]
%Y A329359 Row lengths are A329312.
%Y A329359 Row sums are A070939.
%Y A329359 Positions of rows of length 1 are A275692.
%Y A329359 The non-"co" version is A329314.
%Y A329359 Binary co-Lyndon words are counted by A001037 and ranked by A329318.
%Y A329359 Cf. A059966, A211097, A211100, A328596, A296372, A329313, A329315, A329325, A329357.
%K A329359 nonn,tabf
%O A329359 1,2
%A A329359 _Gus Wiseman_, Nov 12 2019