Constructive proof for the positivity of the orbit polynomial Odn,2(q)

Abstract

The cyclic group $C_n=⟨(12\cdots n)⟩$ acts on the set $(\genfrac{}{}{0ex}{}{[n]}{k})$ of all $k$-subsets of $[n]$. In this action of $C_n$ the number of orbits of size $d$, for $d\mid n$, is
$$O_d^{n,k}=\frac{1}{d}\sum _{\frac{n}{d}\mid s\mid n}\mu \left(\frac{ds}{n}\right)(\genfrac{}{}{0ex}{}{n/s}{k/s}).$$
Stanton and White\cite{sw} generalized the above identity to construct the orbit polynomials

$$O_d^{n,k}(q)=\frac{1}{[d{]}_{q^{n/d}}}\sum _{\frac{n}{d}\mid s\mid n}\mu \left(\frac{ds}{n}\right){\left[\begin{array}{c}n/s\\ k/s\end{array}\right]}_{q^s}$$

and conjectured that $O_d^{n,k}(q)$ have non-negative coefficients. In this paper we give a constructive proof for the positivity of coefficients of the orbit polynomial $O_d^{n,2}(q)$.