Dyck Symmetric Functions and Applications to \(q,t\)-Catalan Polynomials

TL;DR

This paper introduces combinatorial constructions for Dyck sequences, establishing new symmetric function expansions and formulas for the q,t-Catalan polynomial, including tableau and skeleton-based representations.

Contribution

It develops novel bijections, tableau formulas, and skeleton-based expressions for Dyck symmetric functions and the q,t-Catalan polynomial, advancing combinatorial and symmetric function theory.

Findings

Dual Dyck symmetric functions are Schur-positive.

Derived a two-column tableau formula for the q,t-Catalan polynomial.

Organized low-area slices into skeleton-indexed strings with symmetry properties.

Abstract

This paper develops three related combinatorial results for Dyck-type sequences. First, it constructs a row-insertion algorithm for dual Dyck sequences and extends it to Dyck tableaux. This construction gives a weight-preserving bijection between dual Dyck factorizations and pairs consisting of a Dyck tableau and a semistandard Young tableau of the same shape. As a consequence, the associated dual Dyck symmetric functions are Schur-positive, and the corresponding affine Dyck symmetric functions have the conjugate-shape Schur expansion. Second, it applies these Dyck symmetric functions to the \(q,t\)-Catalan polynomial. It gives a two-column tableau formula for \(C_n(q,t)\), expressing it as a sum over Dyck \(m\)-skeletons and at-most-two-column Dyck tableaux with summands involving two-variable Schur functions. Third, it develops a Dyck-skeleton formula for the deficit range \(\defc\le 2n-8\). Full and special Dyck skeletons, together with local \(\mathrm{East}\), \(\mathrm{West}\), \(\mathrm{up}\), and \(\mathrm{down}\) moves, organize the low-area half of each low-deficit slice into skeleton-indexed strings. The \(q,t\)-symmetry of \(C_n(q,t)\) supplies the complementary high-area half in the resulting interval formula.