We consider polynomials of the form $\operatorname{s}_\lambda(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\lambda$ is an integer partition, $\operatorname{s}_\lambda$ is the Schur polynomial associated to $\lambda$, and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated $\varkappa_j$ times. We represent $\operatorname{s}_\lambda(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$ as a quotient whose the denominator is the determinant of the confluent Vandermonde matrix, and the numerator is the determinant of some generalized confluent Vandermonde matrix. We give three algebraic proofs of this formula.

翻译：本文考虑形如$\operatorname{s}_\lambda(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$的多项式，其中$\lambda$为整数分拆，$\operatorname{s}_\lambda$是与$\lambda$关联的Schur多项式，而$y_j^{[\varkappa_j]}$表示$y_j$重复$\varkappa_j$次。我们将$\operatorname{s}_\lambda(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$表示为商式，其分母为合流范德蒙德矩阵的行列式，分子为某个广义合流范德蒙德矩阵的行列式。我们给出了该公式的三个代数证明。