  The goal of this monograph is to study the indicator function for a set of
permutations mapping one finite sequence of positive integers to another from a
representation theoretic, combinatorial and probabilistic perspective. The
degree of a function of permutations is the size of the largest pair of
sequences required when expressing it as a linear combination of these
indicators. This notion of degree, implicit in work of Diaconis, is critical
for many applications of representation theory to extremal combinatorics,
machine learning, probability and statistics. We use the term local to indicate
bounded degree and initiate the study of low degree class functions, which
encode probabilistic data for permutation statistics on each cycle type
simultaneously.
  We begin with a self contained treatment of for functions of permutations,
developing its theory using language familiar to enumerative and algebraic
combinatorialists. This leads naturally to a novel basis for symmetric
functions we call the path power sum symmetric functions. The most technically
challenging part of our work is the path Murnaghan-Nakayama formula, which
expands path power sums into Schur functions. By combining the the path
Murnaghan-Nakayam formula with the classical theory of character polynomials,
one obtains a structural characterization for moments for permutation
statistics conditioning on cycle type. We then analyze asymptotic properties of
these moments. In doing so, we introduce the novel family of regular
permutation statistics, which include almost all reasonable weighted pattern
counting statistics. We show a large family of regular statistics satisfy a law
of large numbers on a given cycle type depending only on the proportion of
fixed points and a have variances depending only on fixed points and two
cycles.
