The Flagged Cauchy Determinant |
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Authors: | William Y C Chen Christian Krattenthaler Arthur L B Yang |
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Institution: | (1) Center for Combinatorics, LPMC, Nankai University, Tianjin, 300071, P.R. China;(2) Institut Girard Desargues, Université Claude Bernard Lyon-I, 21, avenue Claude Bernard, F-69622 Villeurbanne Cedex, France |
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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. |
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Keywords: | Divided difference Cauchy identity Flagged Cauchy determinant Multi-Schur function Nonintersecting lattice paths |
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