A generalized Vandermonde determinant (Q1204335)

The Vandermonde determinant identity is given by \[ \text{det}(x^ j_ i)_{i,j=0,\dots,m}=\prod_{0\leq i<j\leq m}(x_ j-x_ i). \] In this paper two determinantal identities that generalize the Vandermonde determinant identity are established. In the first of the identities the set \(\{0,\dots,m\}\) indexing the rows and columns of the determinant is replaced by an arbitrary finite order ideal in the set of sequences of nonnegative integers which are 0 except for a finite number of terms. In the second the index set is replaced by an arbitrary finite order ideal in the set of all partitions.