  We present conjectured exact expressions for two types of correlations in the
dense O$(n=1)$ loop model on $L\times \infty$ square lattices with periodic
boundary conditions. These are the probability that a point is surrounded by
$m$ loops and the probability that $k$ consecutive points on a row are on the
same or on different loops. The dense O$(n=1)$ loop model is equivalent to the
bond percolation model at the critical point. The former probability can be
interpreted in terms of the bond percolation problem as giving the probability
that a vertex is on a cluster that is surrounded by $\floor{m/2}$ clusters and
$\floor{(m+1)/2}$ dual clusters. The conjectured expression for this
probability involves a binomial determinant that is known to give weighted
enumerations of cyclically symmetric plane partitions and also of certain types
of families of nonintersecting lattice paths. By applying Coulomb gas methods
to the dense O$(n=1)$ loop model, we obtain new conjectures for the asymptotics
of this binomial determinant.
