  We introduce new natural generalizations of the classical descent and
inversion statistics for permutations, called width-$k$ descents and width-$k$
inversions. These variations induce generalizations of the excedance and major
statistics, providing a framework in which the most well-known
equidistributivity results for classical statistics are paralleled. We explore
additional relationships among the statistics providing specific formulas in
certain special cases. Moreover, we explore the behavior of these width-$k$
statistics in the context of pattern avoidance.
