On cyclic compositions of positive integers (Q719661)

Given positive integers \(k,n\) with \(k \leq n,\) it is well known that there are \({n-1} \choose {k-1}\) compositions \(C(n,k)\) of \(n\) into \(k\) parts. Two compositions are related iff they differ only by a shift. The paper studies this equivalence relation. More precisely, if \(Cl(n,k)\) denotes the number of distinct equivalence classes one has \[ Cl(n,k)Cl(n,k+2) \leq Cl(n,k+1)^2 \] for \(4 \leq k+3 \leq n;\) \(Cl(n,k) \equiv {n \choose k} \pmod{2}\) when \(k\) or \(n-1\) are even and \(Cl(n,k) \equiv {{n-1} \choose {k-1}} \pmod{2}\) otherwise. Three questions are asked about a kind of ``characteristic'' polynomial of the sequence \(Cl(n,k)\) divided by different weights. For example the authors proved that for an infinity of odd integers \(n\) the polynomial \[ \frac{\sum_{k=0}^{k=n} Cl(n,k)t^k}{t+1} \] is irreducible.