  The degree of symmetry of a combinatorial object, such as a lattice path, is
a measure of how symmetric the object is. It typically ranges from zero, if the
object is completely asymmetric, to its size, if it is completely symmetric. We
study the behavior of this statistic on Dyck paths and grand Dyck paths, with
symmetry described by reflection along a vertical line through their midpoint;
partitions, with symmetry given by conjugation; and certain compositions
interpreted as bargraphs. We find expressions for the generating functions for
these objects with respect to their degree of symmetry, and their semilength or
semiperimeter, deducing in most cases that, asymptotically, the degree of
symmetry has a Rayleigh or half-normal limiting distribution. The resulting
generating functions are often algebraic, with the notable exception of Dyck
paths, for which we conjecture that it is D-finite (but not algebraic), based
on a functional equation that we obtain using bijections to walks in the plane.
