Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers
Ayumu Hoshino, Jun'ichi Shiraishi

TL;DR
This paper derives explicit formulas and recursion relations for transition matrices involving Macdonald and Koornwinder polynomials of type $C_n$, revealing connections to deformed Catalan numbers and $q$-ballot numbers.
Contribution
It provides new explicit formulas and recursion relations for transition matrices between specialized Macdonald polynomials and monomial symmetric polynomials of type $C_n$, linking them to deformed Catalan and ballot numbers.
Findings
Explicit formula for transition matrix $\\mathcal{C}$ with one-column diagrams.
Recursion relations deforming Catalan triangle or ballot numbers.
Identification of $q$-ballot numbers as Kostka polynomials in specific cases.
Abstract
We present an explicit formula for the transition matrix from the type degeneration of the Koornwinder polynomials with one column diagrams, to the type monomial symmetric polynomials . The entries of the matrix enjoy a set of three term recursion relations, which can be regarded as a -deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type Macdonald polynomials . It is also shown that the -ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials to the Hall-Littlewood polynomials…
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