This material has been published in
Combin. 21 (2005), 51-62,
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The flagged Cauchy determinant
We consider a flagged form
of the Cauchy determinant, for which we provide a combinatorial interpretation
in terms of nonintersecting lattice paths. In combination with the
standard determinant for the enumeration of nonintersecting lattice
paths, we are able to give
a new proof of the Cauchy identity for Schur functions.
Moreover, by choosing
different starting and end points for the lattice paths, we are
led to a lattice path proof of
an identity of Gessel which expresses a Cauchy-like sum
of Schur functions
in terms of the complete symmetric functions.
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