Compositions versus Cyclic Compositions

We prove that the sum of greatest common divisors of parts in all compositions of n equals the sum of lengths of all cyclic compositions of n. The proof highlights structural similarities between the set of compositions of n and the set of cyclic compositions of n. 1. Definitions and Statement A composition of the positive integer n is an ordered partition of n; that is, a vector ( ) r p p ..., , 1 = λ of positive integers , i p called parts, that add up to n. A cyclic composition of n is a composition λ considered only up to a cyclic permutation of its parts (see [4, p. 268]). One can visualize a cyclic composition as a string of numbers arranged clockwise around a circle. Thus, the compositions ( ), 1 , 1 , 2 ( ), 1 , 2 , 1 ( ) 2 , 1 , 1 are all different, but stand for the same cyclic composition of 4, while the compositions ( ) 1 , 2 , 3 and ( ) 2 , 1 , 3 represent two different cyclic compositions of 6. Notation. By convention we represent a cyclic composition by writing the highest composition in its equivalence class using the lexicographical order. Let ( ) n C and ( ) n CC denote the set of compositions of n and the set RODRIGO A. PÉREZ 42 of cyclic compositions of n, respectively. For a (cyclic) composition λ, define ( ) λ gcd as the greatest common divisor of all the parts in λ, and ( ) λ r as the total number of parts in λ; i.e., the length of λ. For , 6 ..., , 1 = n the sum ( ) ( ) ∑ ∈ λ λ n r CC returns the values 1, 3, 6, 12, 20, 42. A search in the OEIS [3] returns sequence A034738, the Dirichlet convolution of 1 2 − n with Euler’s totient function ( ): n φ ( ) ∑ | − φ ⋅ n d n d n . 2 1 (1) The OEIS entry includes a claim to the effect that this sequence equals the sum of greatest common divisors of parts in all compositions of n, but the author of that comment confirmed in an email [2] that such identity “probably is not published anywhere”. We will prove more: Theorem 1. For all , 1 ≥ n ( ) ( ) ( ) ( ) ( ) ∑ ∑ ∑ ∈ λ | ∈ λ − λ = φ ⋅ = λ n n d n n r d n C CC . 2 gcd 1 (2) The connection between greatest common divisors of compositions and lengths of cyclic compositions is initially surprising. The underlying phenomenon is a structural similarity between ( ) n C and ( ) n CC that will be highlighted in our proof. The auxiliary arrays ( ), n C ( ) n CC are intended to reflect the analogy between constructing new compositions by scalar multiplication (a composition transformation related to greatest common divisors) and by concatenation (related to lengths). 2. Preliminary Facts The following four facts are completely elementary and well known. We state them here for ease of reference since they are used at various points in the proof of Theorem 1. Fact I. Every divisor d of 1 ≥ n has associated a symmetric divisor . d n It follows that for any function , : C Z+ g COMPOSITIONS VERSUS CYCLIC COMPOSITIONS 43 ( ) ( ) ∑ ∑ | | = n d n d d n g d g . This will be referred to as a change of divisor variable. Fact II. Let { } n d d X divides : | = and consider two arbitrary functions X S f : and , : C X S F × where S is a finite set. The equality