OFFSET
1,2
COMMENTS
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.
These are co-Lyndon compositions, ordered first by sum and then lexicographically. - Gus Wiseman, Nov 15 2019
FORMULA
Row lengths are A059966(n) = number of prime compositions of n.
EXAMPLE
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:
(1) = (1)
(12) = (1)*(2)
(122) = (1)*(2)*(2)
(1221) = (1)*(221)
(12211) = (1)*(2211)
(122112) = (1)*(2)*(2211)
(1221121) = (1)*(221121)
(12211212) = (1)*(2)*(221121)
(122112122) = (1)*(2)*(2)*(221121)
(1221121221) = (1)*(221)*(221121)
(12211212212) = (1)*(2)*(221)*(221121)
(122112122122) = (1)*(2)*(2)*(221)*(221121).
Read as a sequence:
(1), (2), (21), (3), (211), (31), (4), (2111), (221), (311), (32), (41), (5).
Read as a triangle:
(1)
(2)
(21), (3)
(211), (31), (4)
(2111), (221), (311), (32), (41), (5).
Read as a sequence of triangles:
1 2 2 1 2 1 1 2 1 1 1 2 1 1 1 1 2 1 1 1 1 1
3 3 1 2 2 1 2 2 1 1 2 1 2 1 1
4 3 1 1 3 1 1 1 2 2 1 1 1
3 2 3 1 2 2 2 2 1
4 1 3 2 1 3 1 1 1 1
5 4 1 1 3 1 1 2
4 2 3 1 2 1
5 1 3 2 1 1
6 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.
MATHEMATICA
colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
lexsort[f_, c_]:=OrderedQ[PadRight[{f, c}]];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n], colynQ], lexsort], {n, 5}] (* Gus Wiseman, Nov 15 2019 *)
CROSSREFS
The binary version is A329318.
The binary non-"co" version is A102659.
A sequence listing all Lyndon compositions is A294859.
Numbers whose binary expansion is co-Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Binary Lyndon words are A001037.
Lyndon compositions are A059966.
Normal Lyndon words are A060223.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jan 12 2017
STATUS
approved