Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers

We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{(1^r)}(x\mathrm{\mid }a,-a,c,-c\mathrm{\mid }q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric polynomials $m_{(1^r)}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of three term recursion relations, which can be regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type $(C_n,C_n)$ Macdonald polynomials $P^{(C_n,C_n)}{(1^r)}(x,|,b;q,t)= P{(1^r)}\big(x,|,b^{1/2},-b^{1/2},q^{1/2}b^{1/2},-q^{1/2}b^{1/2},|,q,t\big)$. It is also shown that the $q$-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials $P_{(1^r)}^{(C_n,C_n)}(x\mathrm{\mid }q;q,q)$ to the Hall-Littlewood polynomials $P_{(1^r)}^{(C_n,C_n)}(x\mathrm{\mid }t;0,t)$.