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A299070
Regular triangle T(n,k) is the number of compositions of n whose standard factorization into Lyndon words has k distinct factors.
1
1, 2, 0, 3, 1, 0, 5, 3, 0, 0, 7, 9, 0, 0, 0, 13, 17, 2, 0, 0, 0, 19, 39, 6, 0, 0, 0, 0, 35, 72, 21, 0, 0, 0, 0, 0, 59, 141, 55, 1, 0, 0, 0, 0, 0, 107, 266, 132, 7, 0, 0, 0, 0, 0, 0, 187, 511, 300, 26, 0, 0, 0, 0, 0, 0, 0
OFFSET
1,2
COMMENTS
Row sums are 2^(n-1). First column is A008965. A version without the zeros is A299072.
EXAMPLE
Triangle begins:
1
2 0
3 1 0
5 3 0 0
7 9 0 0 0
13 17 2 0 0 0
19 39 6 0 0 0 0
35 72 21 0 0 0 0 0
59 141 55 1 0 0 0 0 0
107 266 132 7 0 0 0 0 0 0
187 511 300 26 0 0 0 0 0 0 0.
The a(5,2) = 9 compositions are (41), (32), (311), (131), (221), (212), (2111), (1211), (1121) with factorizations
(41) = (4) * (1)
(32) = (3) * (2)
(311) = (3) * (1)^2
(131) = (13) * (1)
(221) = (2)^2 * (1)
(212) = (2) * (12)
(2111) = (2) * (1)^3
(1211) = (12) * (1)^2
(1121) = (112) * (1).
MATHEMATICA
LyndonQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And]&&Array[RotateRight[q, #]&, Length[q], 1, UnsameQ];
qit[q_]:=If[#===Length[q], {q}, Prepend[qit[Drop[q, #]], Take[q, #]]]&[Max@@Select[Range[Length[q]], LyndonQ[Take[q, #]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[Union[qit[#]]]===k&]], {n, 11}, {k, n}]
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Feb 01 2018
STATUS
approved