Let $A(n,r;3)$ be the total weight of the alternating sign matrices of order $n$ whose sole `1 of the first row is at the $r^{th}$ column and the weight of an individual matrix is $3^k$ if it has $k$ entries equal to -1. Define the sequence of the generating functions $G_n(t)=sum_{r=1}^n A(n,r;3)t^{r-1}$. Results of two different kind are obtained. On the one hand I made the explicit expression for the even subsequence $G_{2 u}(t)$ in terms of two linear homogeneous second order recurrence in $ u$ (Theorem 1). On the other hand I brought to light the nice connection between the neighbouring functions $G_{2 u+1}(t)$ and $G_{2 u}(t)$ (Theorem 2). The 3-enumeration $A(n;3) equiv G_n(1)$ which was found by Kuperberg is reproduced as well.
تحميل البحث