We evaluate four families of determinants of matrices, where the entries are
sums or differences of generating functions for paths consisting of up-steps,
down-steps and level steps. By specialisation, these determinant evaluations
have numerous corollaries. In particular, they cover numerous determinant
evaluations of combinatorial numbers - most notably of Catalan, ballot, and of
Motzkin numbers - that appeared previously in the literature.

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in
the plane, and let $\Cal H$ be the half plane to the left of $\ell$. We
consider lozenge tilings of $\Cal H$ that have a triangular gap of side-length
two and in which $\ell$ is a free boundary - i.e., tiles are allowed to
protrude out half-way across $\ell$. We prove that the correlation function of
this gap near the free boundary has asymptotics $\frac{1}{4\pi r}$,
$r\to\infty$, where $r$ is the distance from the gap to the free boundary.

We develop a new setting for the exponential principle in the context of
multisort species, where indecomposable objects are generated intrinsically
instead of being given in advance. Our approach uses the language of functors
and natural transformations (composition operators), and we show that, somewhat
surprisingly, a single axiom for the composition already suffices to guarantee
validity of the exponential formula. We provide various illustrations of our
theory, among which are applications to the enumeration of (semi-)magic squares
and (higher-dimensional) cubes.

We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf
G}(z)/{\bf F}(z))$ are integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z)
{\bf F}(z)$ are specific solutions of certain hypergeometric differential
equations with maximal unipotent monodromy at $z=0$. We also address the
question of finding the largest integer $u$ such that the Taylor coefficients
of $(z ^{-1}{\bf q}(z))^{1/u}$ are still integers.

For the Moebius sphere $\S^{q,p}$, we confirm the recursive formulas for
GJMS-operators and $Q$-curvatures formulated by the first author in "On
conformally covariant powers of the Laplacian" {arXiv:0905.3992}