A328669 Number of Lyndon compositions of n where every pair of adjacent parts (including the last with the first) is relatively prime.

1, 0, 1, 2, 4, 6, 11, 18, 31, 52, 93, 157, 278, 479, 846, 1486, 2646, 4675, 8348, 14864, 26629, 47699, 85777, 154289, 278317, 502436, 908879, 1645712, 2984545, 5417742, 9847188, 17914493, 32625522, 59467892, 108493133, 198089609, 361965237, 661883230, 1211161990

Offset: 1

Author

Gus Wiseman, Oct 26 2019

Keywords

• nonn

Comments

A Lyndon composition of n is a finite sequence of positive integers summing to n that is lexicographically strictly less than all of its cyclic rotations.

Examples

			The a(1) = 1 through a(8) = 18 Lyndon compositions (empty column not shown):
  (1)  (12)  (13)   (14)    (15)     (16)      (17)
             (112)  (23)    (114)    (25)      (35)
                    (113)   (123)    (34)      (116)
                    (1112)  (132)    (115)     (125)
                            (1113)   (1114)    (134)
                            (11112)  (1123)    (143)
                                     (1132)    (152)
                                     (1213)    (1115)
                                     (11113)   (1214)
                                     (11212)   (1232)
                                     (111112)  (11114)
                                               (11123)
                                               (11132)
                                               (11213)
                                               (11312)
                                               (111113)
                                               (111212)
                                               (1111112)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

The non-Lyndon version is A328609 or A318748 (with singletons).

The non-Lyndon non-circular version is A167606.

The version with singletons is A318745.

The necklace case is A328597 or A318728 (with singletons).

The aperiodic case is A328670.

Lyndon compositions are A059966, with relatively prime case A318731.

Cf. A000031, A000740, A008965, A328172, A328601, A328602.

Programs

Mathematica

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],aperQ[#]&&neckQ[#]&&And@@CoprimeQ@@@Partition[#,2,1,1]&]],{n,10}]

PARI

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->gcd(i, j)==1))); vector(n, n, sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

Formula

a(n > 1) = A318745(n) - 1.