Christian Krattenthaler

Non-crossing two-rowed arrays and summations for Schur functions

(15 pages)

Abstract. In the first part of this paper (sections 1,2) we give combinatorial proofs for determinantal formulas for sums of Schur functions ``in a strip" that were originally obtained by Gessel, respectively Goulden, using algebraic methods. The combinatorial analysis involves certain families of two-rowed arrays, asymmetric variations of Sagan and Stanley's skew Knuth-correspondence, and variations of one of Burge's correspondences. In the third section we specialize the parameters in these determinants to compute norm generating functions for tableaux in a strip. In case we can get rid of the determinant we obtain multifold summations that are basic hypergeometric series for Ar and Cr respectively. In some cases these sums can be evaluated. Thus in particular, an alternative proof for refinements of the Bender-Knuth and MacMahon (ex-)Conjectures, which were first obtained in another paper by the author, is provided. Although there are some parallels with the original proof, perhaps this proof is easier accessible. Finally, in section 4, we record further applications of our methods to the enumeration of paths with respect to weighted turns.

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