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

In the rst part of this paper (sections 1,2) we give combinatorial proofs for de-terminantal 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 A r and C r respectively. In some cases these sums can be evaluated. Thus in particular, an alternative proof for reenements of the Bender-Knuth and MacMahon (ex-)Conjectures, which were rst 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. 1. Generating functions for non-crossing two-rowed arrays. We consider two-rowed arrays P = (p j q) of the form where a; k; b are some nonnegative integers and where the entries p i ; q i are positive integers such that both rows of the array are weakly increasing. (To be precise, if k = 0, i.e. the \middle part" of the array is empty, for a c minfa; bg we also allow the entries p ?1 ; : : : ; p ?c and q ?1 ; : : : ; q ?c to be \empty".) We say that P is of the type (a; b) and of the shape (a; k; b). If both rows of P are strictly increasing then we call P a strict two-rowed array. Given an array P 1 = (p (1) j q (1)) of the shape (a 1 ; k 1 ; b 1) and an array P 2 = (p (2) j q (2)) of the shape (a 2 ; k 2 ; b 2), we say that P 1 dominates (resp. strictly dominates) P 2 if the following three conditions hold: (D1) a 1 a 2 and p (1) l p (2) l (resp. p (1) l < p (2) l) for all l = ?1;?2;:::;?minfa 1 ; a 2 g. (By convention, these inequalities …