A299070 - OEIS

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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, 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

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OFFSET

1,2

COMMENTS

Row sums are 2^(n-1). First column is A008965. A version without the zeros is A299072.

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

Triangle begins:

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}]

CROSSREFS

Cf. A001045, A001221, A008965, A059966, A116608, A146289, A185700, A296373, A299072.

Sequence in context: A321878 A225084 A238345 * A209599 A238347 A263322

Adjacent sequences: A299067 A299068 A299069 * A299071 A299072 A299073

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Feb 01 2018

STATUS

approved