Abstract
We prove two determinantal identities that generalize the Vandermondedeterminant identity \(\det (x_i^j )_{i,j = 0, \ldots ,m} = \prod\limits_{0 \leqslant i < j \leqslant m} {(x_j - x_i )}\). In the first of our identities the set {0, ..., m} indexing the rows and columns of thedeterminant is replaced by an arbitrary finite order ideal in the set ofsequences of nonnegative integers which are 0 except for a finite numberof components. In the second the index set is replaced by an arbitraryfinite order ideal in the set of all partitions.
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Reference
E.A. Bender, R.A. Coley, D.P. Robbins, and H. Rumsey, Jr., “Enumeration of subspaces by dimension sequence,” J. Combin. Theory Ser. A, 59 (1992), 1–11.
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Buck, M.W., Coley, R.A. & Robbins, D.P. A Generalized Vandermonde Determinant. Journal of Algebraic Combinatorics 1, 105–109 (1992). https://doi.org/10.1023/A:1022468019197
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DOI: https://doi.org/10.1023/A:1022468019197