[Paper Review] Parabolic Catalan numbers count flagged Schur functions; Convexity of tableau sets for Demazure characters

Robert A. Proctor, Matthew J. Willis|arXiv (Cornell University)|Dec 19, 2016

Advanced Combinatorial Mathematics3 citations

TL;DR

This paper introduces parabolic Catalan numbers as the count of 312-avoiding R-permutations—ordered partitions of {1,...,n} with block sizes derived from a subset R ⊆ [n−1]. It establishes that these numbers enumerate distinct flagged Schur functions and characterize when Demazure character tableau sets are convex, showing that convexity holds precisely for 312-avoiding R-permutations. The key contribution is a deep combinatorial coincidence: flagged Schur functions and Demazure characters coincide exactly when the indexing permutation is 312-avoiding, with underlying tableau sets matching at the level of integer lattice convexity.

ABSTRACT

Shuffles are n-multipermutations with suit multiplicities given by a subset R of {1,..,n-1}. Their inverses are ordered partitions of {1,..,n} whose block sizes derive from R. These "R-permutations" depict the min length coset reps for the quotient of S_n by the parabolic subgroup W_J, with J the complement of R. We refer to those that blockwise avoid the pattern 312 as "312-avoiding R-permutations" and define the "parabolic R-Catalan number" to be the number of them. Let lambda be a partition of N with at most n parts whose set of shape column lengths less than n is R. We show that the number of flagged Schur functions formed on the shape of lambda is this parabolic R-Catalan number, and list over a dozen other phenomena that are enumerated by it. Let pi be an R-permutation. We view the Demazure character (key polynomial) indexed by (lambda,pi) as the sum of the content weight monomials for our "pi-Demazure" semistandard tableaux of shape lambda with entries from {1,..,n}. We show that the set of these tableaux is convex in Z^N if and only if pi is a 312-avoiding R-permutation. A flagged Schur function is the sum of the content weight monomials for the semistandard tableaux of shape lambda whose entries are row-wise bounded by a given weakly increasing n-tuple. We consider general row bound sums for which the bounds may be any n-tuple. Reiner and Shimozono and then Postnikov and Stanley obtained results concerning coincidences between flagged Schur functions and Demazure characters: when lambda is strict, the flagged Schur functions exactly coincide with the 312-avoiding Demazure characters. For general lambda we introduce precise indexing sets of n-tuple bounds for the row bound sums. This and our convexity results are used to sharpen their coincidence results, to extend them to general row bound sums, and to show they hold at the deeper level of coinciding underlying tableau sets.

Motivation & Objective

• To define and enumerate a new class of combinatorial objects—R-parabolic Catalan numbers—via 312-avoiding R-permutations.
• To establish a precise correspondence between flagged Schur functions and Demazure characters by identifying the conditions under which their underlying tableau sets coincide.
• To characterize the convexity of Demazure tableau sets in ℤ^N and show it holds if and only if the indexing R-permutation is 312-avoiding.
• To refine the indexing of row-bound sums for flagged Schur functions using 'gapless R-tuples' that bijectively correspond to 312-avoiding R-permutations.
• To extend and sharpen prior results by Reiner, Shimozono, Postnikov, and Stanley on coincidences between flagged Schur functions and Demazure characters, now at the level of tableau sets rather than just polynomials.

Proposed method

• Define R-permutations as ordered partitions of {1,...,n} whose block sizes are derived from a subset R ⊆ [n−1], corresponding to minimal length coset representatives in S_n / W_J where J = [n−1] ∓ R.
• Introduce 'R-312-avoidance' for R-permutations, generalizing classical 312-pattern avoidance in permutations, and define the parabolic R-Catalan number as the count of such permutations.
• Use the 'R-ranking' map to construct a bijection between 312-avoiding R-permutations and 'gapless R-tuples', which index the most efficient row-bound sums for flagged Schur functions.
• Define 'π-Demazure' semistandard Young tableaux as those with entries from [n] and content weight monomials summing to the Demazure character indexed by (λ, π), and prove their set is convex in ℤ^N iff π is 312-avoiding.
• Establish that flagged Schur functions and Demazure characters coincide not only as polynomials but also as sets of tableaux when the indexing permutation is 312-avoiding, using convexity and indexing refinements.
• Apply the Gessel-Viennot determinant formula to enumerate distinct flagged Schur functions and Demazure characters, showing their number equals the parabolic R-Catalan number.

Experimental results

Research questions

• RQ1 What is the combinatorial interpretation of the parabolic R-Catalan number, and how does it generalize the classical Catalan number?
• RQ2 Under what conditions do flagged Schur functions and Demazure characters coincide as polynomials, and can this coincidence be strengthened to a coincidence of their underlying tableau sets?
• RQ3 When is the set of π-Demazure tableaux for a given shape λ convex in ℤ^N, and how does this relate to pattern avoidance in the indexing permutation π?
• RQ4 What is the most efficient indexing scheme for row-bound sums that yield distinct flagged Schur functions, and how is it related to 312-avoiding R-permutations?
• RQ5 How do the results of Reiner, Shimozono, Postnikov, and Stanley on coincidences between flagged Schur functions and Demazure characters extend when considering general row-bound sums and convexity of tableau sets?

Key findings

• The number of distinct flagged Schur functions on a given shape λ with column lengths less than n forming the set R is exactly the parabolic R-Catalan number, which counts 312-avoiding R-permutations.
• The set of π-Demazure tableaux is convex in ℤ^N if and only if the R-permutation π is 312-avoiding, establishing a deep structural link between pattern avoidance and convexity.
• Flagged Schur functions and Demazure characters coincide as polynomials if and only if the indexing permutation is 312-avoiding, and this coincidence extends to the level of identical tableau sets.
• The most efficient indexing for row-bound sums that yield distinct flagged Schur functions corresponds bijectively to 312-avoiding R-permutations via the 'R-ranking' map, producing 'gapless R-tuples'.
• The total parabolic Catalan number C_n^Σ counts not only 312-avoiding R-permutations but also rightmost clump deleting chains, gapless keys with distinct column lengths < n, and Schubert varieties in SL(n)/P_J with convex Demazure tableaux.
• The paper provides a unifying framework that extends and sharpens earlier results by Reiner, Shimozono, Postnikov, and Stanley, showing that their coincidences are not coincidental but arise from a deeper combinatorial equivalence rooted in convexity and pattern avoidance.