Row Bounds Needed To Justifiably Express Flagged Schur Functions With Gessel-Viennot Determinants

Let lambda be a partition with no more than n parts. Let beta be a weakly increasing n-tuple with entries from {1, ..., n}. The flagged Schur function in the variables x(1), ..., x(n) that is indexed by lambda and beta has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape lambda whose values are row-wise bounded by the entries of beta. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by lambda and beta; this could be done since the pair (lambda, beta) satisfied their "nonpermutable" condition for the sequence of terminals of an n-tuple of lattice paths that they used to model the tableaux. We generalize flagged Schur functions by dropping the requirement that beta be weakly increasing. Then for each lambda we give a condition on the entries of beta for the pair (lambda, beta) to be nonpermutable that is both necessary and sufficient. When the parts of lambda are not distinct there will be multiple row bound n-tuples beta that will produce the same set of tableaux. We accordingly group the bounding beta into equivalence classes and identify the most efficient beta in each class for the determinant computation. We recently showed that many other sets of objects that are indexed by n and lambda are enumerated by the number of these efficient n-tuples. We called these counts "parabolic Catalan numbers". It is noted that the GL(n) Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants.