  We explore the notion of degree of asymmetry for integer sequences and
related combinatorial objects. The degree of asymmetry is a new combinatorial
statistic that measures how far an object is from being symmetric. We define
this notion for compositions, words, matchings, binary trees and permutations,
we find generating functions enumerating these objects with respect to their
degree of asymmetry, and we describe the limiting distribution of this
statistic in each case.
