Non-crossing two-rowed arrays and summations for Schur
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
respectively Goulden, using
algebraic methods. The combinatorial analysis involves
certain families of two-rowed arrays, asymmetric
variations of Sagan
and Stanley's skew
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
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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