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

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.