- COMBINATORIAL PROOF FOR THE POSITIVITY OF THE ORBIT POLYNOMIAL $O^{n,3}_d(q)$
- ㆍ 저자명
- Lee. Jae-Jin
- ㆍ 간행물명
- Journal of applied mathematics & informatics
- ㆍ 권/호정보
- 2012년|30권 3호|pp.455-462 (8 pages)
- ㆍ 발행정보
- 한국전산응용수학회
- ㆍ 파일정보
- 정기간행물| PDF텍스트
- ㆍ 주제분야
- 기타
The cyclic group $Cn={langle}(12{cdots}n){ angle}$ acts on the set ($^{[n]}_k$) of all $k$-subsets of [$n$]. In this action of $C_n$ the number of orbits of size $d$, for $d|n$, is $$O^{n,k}_d=frac{1}{d}sum_{frac{n}{d}|s|n}{mu}(frac{ds}{n})(^{n/s}_{k/s})$$. Stanton and White[7] generalized the above identity to construct the orbit polynomials $$O^{n,k}_d(q)=frac{1}{[d]_{q^{n/d}}}sum_{frac{n}{d}|s|n}{mu}(frac{ds}{n})[^{n/s}_{k/s}]{_q}^s$$ and conjectured that $O^{n,k}_d(q)$ have non-negative coefficients. In this paper we give a combinatorial proof for the positivity of coefficients of the orbit polynomial $O^{n,3}_d(q)$.