Constructive proof for the positivity of the orbit polynomial $O_d^{n,2}(q)$
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The cyclic group $C_n=\langle (12\cdots n)\rangle$ acts on the set $\binom {[n]}k$ of all $k$-subsets of $[n]$. In this action of $C_n$ the number of orbits of size $d$, for $d\mid n$, is $$ O_d^{n,k}=\frac 1d\sum_{\frac nd\mid s\mid n}\mu\left(\frac{ds}n\right)\binom{n/s}{k/s}. $$ Stanton and White\cite{sw} generalized the above identity to construct the orbit polynomials $$ O_d^{n,k}(q)=\frac{1}{[d]_{q^{n/d}}}\sum_{\frac nd\mid s\mid n} \mu\left(\frac{ds}n\right) \left[ \ \begin{matrix} n/s \\ k/s \end{matrix} \ \right]_{q^s} $$ 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 $O_d^{n,2}(q)$.
Abstract
The cyclic group $C_n=\langle (12\cdots n)\rangle$ acts on the set $\binom {[n]}k$ of all $k$-subsets of $[n]$. In this action of $C_n$ the number of orbits of size $d$, for $d\mid n$, is $$ O_d^{n,k}=\frac 1d\sum_{\frac nd\mid s\mid n}\mu\left(\frac{ds}n\right)\binom{n/s}{k/s}. $$ Stanton and White\cite{sw} generalized the above identity to construct the orbit polynomials $$ O_d^{n,k}(q)=\frac{1}{[d]_{q^{n/d}}}\sum_{\frac nd\mid s\mid n} \mu\left(\frac{ds}n\right) \left[ \ \begin{matrix} n/s \\ k/s \end{matrix} \ \right]_{q^s} $$ 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 $O_d^{n,2}(q)$.
참고문헌 (7)
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