Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers

TL;DR

This paper establishes a convexity property of tableau sets associated with type A Demazure characters, linking R-312-avoidance in R-permutations to geometric and combinatorial structures, and introduces R-parabolic Catalan numbers.

Contribution

It proves that the set of Demazure tableaux is convex if and only if the permutation is R-312-avoiding, connecting combinatorial avoidance patterns with geometric convexity.

Findings

Convexity of tableau sets characterized by R-312-avoidance.

R-parabolic Catalan numbers count R-312-avoiding permutations.

Reformulation of special permutations as R-rightmost clump deleting chains.

Abstract

This is the first of three papers that develop structures which are counted by a "parabolic" generalization of Catalan numbers. Fix a subset R of {1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are determined by R. These are the "inverses" of (parabolic) multipermutations whose multiplicities are determined by R. The standard forms of the ordered partitions are refered to as "R-permutations". The notion of 312-avoidance is extended from permutations to R-permutations. Let lambda be a partition of N such that the set of column lengths in its shape is R or R union {n}. Fix an R-permutation pi. The type A Demazure character (key polynomial) in x_1, .., x_n that is indexed by lambda and pi can be described as the sum of the weight monomials for some of the semistandard Young tableau of shape lambda that are used to describe the Schur function indexed by lambda. Descriptions of these "Demazure" tableaux developed by the authors in earlier papers are used to prove that the set of these tableaux is convex in Z^N if and only if pi is R-312-avoiding if and only if the tableau set is the entire principal ideal generated by the key of pi. These papers were inspired by results of Reiner and Shimozono and by Postnikov and Stanley concerning coincidences between Demazure characters and flagged Schur functions. This convexity result is used in the next paper to deepen those results from the level of polynomials to the level of tableau sets. The R-parabolic Catalan number is defined to be the number of R-312-avoiding permutations. These special R-permutations are reformulated as "R-rightmost clump deleting" chains of subsets of {1,..,n} and as "gapless R-tuples"; the latter n-tuples arise in multiple contexts in these papers.