A365859 Number of self-dual cyclic n-color compositions.

1, 1, 2, 1, 3, 2, 5, 1, 10, 3, 19, 2, 41, 5, 94, 1, 211, 10, 493, 3, 1170, 19, 2787, 2, 6713, 41, 16274, 5, 39651, 94, 97109, 1, 238838, 211, 589527, 10, 1459961, 493, 3626242, 3, 9030451, 1170, 22542397, 19, 56393862, 2787, 141358275, 2, 354975433, 6713, 892893262, 41, 2249412291, 16274, 5674891017

Offset: 1

Author

Joshua P. Bowman, Sep 20 2023

Keywords

• nonn

Comments

A cyclic composition is a sum in which the order of the parts is considered up to cyclic permutation. In other words, it is the collection of components remaining in the cycle graph C_n on n vertices when one or more edges are removed, and rotations are considered equivalent. In an n-color composition, each part of size k is assigned one of k "colors" which may be represented graphically by marking one vertex in the part. The dual of a cyclic n-color composition is obtained by switching the roles of edges and vertices in C_n, then removing each edge that came from a previously marked vertex while marking each vertex that came from a previously removed edge. A cyclic n-color composition is self-dual if it is invariant under this process.

a(n) is also the number of cyclic compositions of A000265(n) into odd parts.

This sequence is self-similar; removing all odd-indexed terms results in the same sequence.

Examples

			Every power of 2 has only one self-dual cyclic n-color composition, which has all parts of size 1.
The self-dual cyclic n-color compositions of 5 are 1_1+1_1+1_1+1_1+1_1, 1_1+2_2+2_1, and 5_3, where the subscript indicates the color of the part, or which vertex is marked within the part.
		

Links

• A. K. Agarwal, n-colour compositions, Indian J. Pure Appl. Math. 31 (11) (2000), 1421-1427.
• Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 33.
• Meghann Moriah Gibson, Daniel Gray, and Hua Wang, Combinatorics of n-color cyclic compositions, Discrete Mathematics 341 (2018), 3209-3226.
• Brian Hopkins, Jesús Sistos Barrón, and Hua Wang, Conjugating cyclic n-color compositions, Utilitas Mathematica (2025) Vol. 122, 53-64. See p. 61.
• Jesus Omar Sistos Barron, Counting Conjugates of Colored Compositions, Honors College Thesis, Georgia Southern Univ. (2024), No. 985. See p. 25.

Crossrefs

Cf. A000204 (Lucas), A000265, A032189, A032198, A365857, A365858.

Programs

PARI

my(N=66,x='x+O('x^N)); Vec( sum(k=1,N, eulerphi(2*k)/(2*k) * log((1+x^k-x^(2*k))/(1-x^k-x^(2*k))) ) )  \\ Joerg Arndt, Sep 21 2023

Python

from sympy import totient, lucas, divisors
def A365859(n):
    m = n>>(~n&n-1).bit_length()
    return sum(totient(k)*lucas(m//k) for k in divisors(m,generator=True))//m # Chai Wah Wu, Sep 23 2023

Formula

G.f.: Sum_{k>=1} phi(2*k)/(2*k) * log((1+x^k-x^(2*k))/(1-x^k-x^(2*k))).

a(n) = (1/(b(n)))*[Sum_{k divides A000265(n)} phi(k)*lucas(b(n)/k)], where b(n) = A000265(n) and lucas(n) = A000204(n).

a(n) = 2*A365857(n) - A032198(n).