The Flagged Cauchy Determinant

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.

[1]  Ira M. Gessel,et al.  Symmetric functions and P-recursiveness , 1990, J. Comb. Theory, Ser. A.

[2]  Christian Krattenthaler,et al.  Non-crossing two-rowed arrays and summations for Schur functions , 1993 .

[3]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[4]  Symplectic Shifted Tableaux and Deformations of Weyl's Denominator Formula for sp(2n) , 2002 .

[5]  A. Lascoux.  Symmetric Functions and Combinatorial Operators on Polynomials , 2003 .

[6]  Ian P. Goulden,et al.  Planar decompositions of tableaux and Schur function determinants , 1995, Eur. J. Comb..

[7]  Mihai Ciucu,et al.  Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole , 2001, J. Comb. Theory, Ser. A.

[8]  I. Gessel,et al.  Binomial Determinants, Paths, and Hook Length Formulae , 1985 .

[9]  Sergey Fomin,et al.  Schensted Algorithms for Dual Graded Graphs , 1995 .

[10]  Michelle L. Wachs,et al.  Flagged Schur Functions, Schubert Polynomials, and Symmetrizing Operators , 1985, J. Comb. Theory, Ser. A.

[11]  Christian Krattenthaler,et al.  Lattice Path Proofs for Determinantal Formulas for Symplectic and Orthogonal Characters , 1997, J. Comb. Theory, Ser. A.

[12]  Angèle M. Hamel,et al.  Pfaffians and Determinants for Schur Q-Functions , 1996, J. Comb. Theory, Ser. A.

[13]  Samuel Karlin,et al.  COINCIDENT PROPERTIES OF BIRTH AND DEATH PROCESSES , 1959 .

[14]  John R. Stembridge,et al.  Nonintersecting Paths, Pfaffians, and Plane Partitions , 1990 .

[15]  William Y. C. Chen,et al.  The Flagged Double Schur Function , 2002 .

[16]  David M. Bressoud,et al.  Determinental Formulae for Complete Symmetric Functions , 1992, J. Comb. Theory, Ser. A.

[17]  Ira M. Gessel,et al.  Determinants, Paths, and Plane Partitions , 1989 .

[18]  Bruce E. Sagan,et al.  Robinson-schensted algorithms for skew tableaux , 1990, J. Comb. Theory A.

[19]  Curtis Greene,et al.  A New Tableau Representation for Supersymmetric Schur Functions , 1994 .

[20]  Tom Roby,et al.  Applications and extensions of Fomin's generalization of the Robinson-Schensted correspondence to differential posets , 1991 .

[21]  A. Hamel.  Determinantal Forms for Symplectic and Orthogonal Schur Functions , 1997, Canadian Journal of Mathematics.

[22]  C. Krattenthaler.  Oscillating tableaux and nonintersecting lattice paths , 1996 .

[23]  F. Brenti.  Determinants of Super-Schur Functions, Lattice Paths, and Dotted Plane Partitions , 1993 .

[24]  Lattice path proof of the ribbon determinant formula for Schur functions , 1991 .

[25]  B. Lindström.  On the Vector Representations of Induced Matroids , 1973 .