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

dmtcs:6632 - Discrete Mathematics & Theoretical Computer Science, April 23, 2021, vol. 23 no. 1 -
Row bounds needed to justifiably express flagged Schur functions with Gessel-Viennot determinantsArticle

Authors: 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.

    Volume: vol. 23 no. 1
    Section: Combinatorics
    Published on: April 23, 2021
    Accepted on: April 1, 2021
    Submitted on: July 10, 2020
    Keywords: Mathematics - Combinatorics,05E05, 05A19

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