范德蒙德行列式、舒尔函数和递归公式的幂
Powers of the Vandermonde determinant, Schur Functions, and recursive formulas
摘要：
由于范德蒙德行列式的每一个偶次幂都是一个对称多项式，所以我们想用舒尔函数的基来理解它的分解。我们研究了分解中系数的几个组合性质。特别是，如果$ł$的Young图是由$ł$的Young图加上一个Tetris型得到的，则我们给出了$n + 1$变量的Schur函数$s_{\m}$在Vandermonde行列式的偶次幂分解中的系数的递推公式，它是由$n$变量的Schur函数$s_ł$在Vandermonde行列式的同偶次幂分解中的系数给出的形状向顶部或向左。一个扩展的摘要包含了这里提出的结果的陈述，出现在FPSAC11的会议记录中。
英文摘要：
Since every even power of the Vandermonde determinant is a symmetric polynomial, we want to understand its decomposition in terms of the basis of Schur functions. We investigate several combinatorial properties of the coefficients in the decomposition. In particular, we give recursive formulas for the coefficient of the Schur function $s_{\m}$ in the decomposition of an even power of the Vandermonde determinant in $n + 1$ variables in terms of the coefficient of the Schur function $s_ł$ in the decomposition of the same even power of the Vandermonde determinant in $n$ variables if the Young diagram of $\m$ is obtained from the Young diagram of $ł$ by adding a tetris type shape to the top or to the left. An extended abstract containing the statement of the results presented here appeared in the Proceedings of FPSAC11
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