The Flagged Cauchy Determinant

William Y. C. Chen,   C. Krattenthaler   and   Arthur L. B. Yang

  Abstract: 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.

  AMS Classification: 05E05, 05A15.

  Keywords:  Divided difference, Cauchy theorem, flagged Cauchy determinant, multi-Schur function, lattice paths.

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