OFFSET
1,3
COMMENTS
We consider cyclic compositions (necklaces) as equivalence classes of compositions that can be obtained from each other by a cyclic shift. A cyclic composition is called Sommerville symmetrical (on a symmetric necklace) if its equivalence class contains at least one palindromic composition (type I) or a composition that becomes a palindromic composition if we remove the first part (type II). A composition with only one part is a palindromic composition of both types.
The equivalence class of each Sommerville symmetrical cyclic composition that is Carlitz contains exactly two type II palindromic Carlitz compositions (except in the case of a composition with only one part). For example, when n = 8, the equivalence class {(1,2,3,2), (2,3,2,1), (3,2,1,2), (2,1,2,3)} represents a Sommerville symmetrical cyclic composition of n = 8 that is Carlitz, but only two of the compositions in the set, i.e., (1,2,3,2) and (3,2,1,2), are type II palindromic.
LINKS
Petros Hadjicostas, Cyclic, dihedral and symmetrical Carlitz compositions of a positive integer, Journal of Integer Sequences, 20 (2017), Article 17.8.5.
Petros Hadjicostas, Generalized colored circular palindromic compositions, Moscow Journal of Combinatorics and Number Theory, 9(2) (2020), 173-186.
P. Hadjicostas and L. Zhang, Sommerville's symmetrical cyclic compositions of a positive integer with parts avoiding multiples of an integer, Fibonacci Quarterly 55 (2017), 54-73.
D. M. Y. Sommerville, On certain periodic properties of cyclic compositions of numbers, Proc. London Math. Soc. S2-7(1) (1909), 263-313.
FORMULA
a(n) = (A291941(n) + 1)/2.
G.f.: x/(1 - x) - A(x)/2 + B(x)^2/(2*(1 - A(x)), where A(x) = Sum_{n >= 1} x^(2*n)/(1 + x^(2*n)) and B(x) = Sum_{n >= 1} x^n/(1 + x^(2*n)).
EXAMPLE
For n = 7, there are exactly a(7) = 5 Sommerville symmetrical cyclic compositions (symmetric necklaces) of 7 that are Carlitz: 7, 1+6, 2+5, 3+4, 2+1+3+1. (Note that 1+6 is the same as 6+1, 3+1+2+1 is the same as 2+1+3+1, and so on, because in each case one composition can be obtained from the other by a cyclic shift.)
CROSSREFS
KEYWORD
nonn
AUTHOR
Petros Hadjicostas, Sep 11 2017
EXTENSIONS
More terms from Altug Alkan, Sep 18 2017
STATUS
approved