Row bounds needed to justifiably express flagged Schur functions with
Gessel-Viennot determinants
Authors: Robert A. Proctor ; Matthew J. Willis
NULL##NULL
Robert A. Proctor;Matthew J. Willis
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.