  Many researchers have studied symmetry properties of various Boolean
functions. A class of Boolean functions, called nested canalyzing functions
(NCFs), has been used to model certain biological phenomena. We identify some
interesting relationships between NCFs, symmetric Boolean functions and a
generalization of symmetric Boolean functions, which we call $r$-symmetric
functions (where $r$ is the symmetry level). Using a normalized representation
for NCFs, we develop a characterization of when two variables of an NCF are
symmetric. Using this characterization, we show that the symmetry level of an
NCF $f$ can be easily computed given a standard representation of $f$. We also
present an algorithm for testing whether a given $r$-symmetric function is an
NCF. Further, we show that for any NCF $f$ with $n$ variables, the notion of
strong asymmetry considered in the literature is equivalent to the property
that $f$ is $n$-symmetric. We use this result to derive a closed form
expression for the number of $n$-variable Boolean functions that are NCFs and
strongly asymmetric. We also identify all the Boolean functions that are NCFs
and symmetric.
