Some composition determinants

We compute several parametric determinants in which rows and columns are indexed by compositions, where the entries are either products of binomial coefficients or products of powers. These results generalize previous determinant evaluations due to the first and fourth author [SIAM J. Matrix Anal. Appl. 23 (2001) 459–471] and [A polynomial generalization of the power-compositions determinant, Linear Multilinear Algebra, in press], and they prove two conjectures of the second author [Advanced determinant calculus: a complement, preliminary version].