Parabolic Catalan numbers count efficient inputs for Gessel-Viennot flagged Schur function determinant

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 condition for the sequence of terminals of an n-tuple of certain lattice paths that they used to model the tableaux. We generalize the notion of flagged Schur function by dropping the requirement that lambda be weakly increasing. Then we give a condition on the entries of lambda and 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 that will produce the same polynomial via the sum of tableau weights construction on lambda. We accordingly group the bounding n-tuples into equivalence classes and identify the most efficient n-tuple in each class for the determinant computation. We have recently shown that many other sets of objects that are indexed by n and lambda are enumerated by the number of these efficient n-tuples. It is noted that the GL(n) Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants.