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Home > Vol 1, No 2 (1992)

Marshall W. Buck , Raymond A. Coley and David P. Robbins

We prove two determinantal identities that generalize the Vandermondedeterminant identity det( x _i ^j ) _{i, j = 0, \frac{1}{4} , m} = Õ _{0 \leqslant i < j \leqslant m} ( x _j - x _i ) \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.

1. E.A. Bender, R.A. Coley, D.P. Robbins, and H. Rumsey, Jr., "Enumeration of subspaces by dimension sequence,"
7. Combin. Theory Ser. A, 59 (1992), 1-11.