Dyck Symmetric Functions and Applications to $q,t$-Catalan Polynomials

Graham Hawkes

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arXiv · 2605.13003

Dyck Symmetric Functions and Applications to $q,t$-Catalan Polynomials

Graham Hawkes

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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.

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BibTeX RIS

Graham Hawkes. 2026-05-13. Dyck Symmetric Functions and Applications to $q,t$-Catalan Polynomials. https://arxiv.org/abs/2605.13003

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