  Inspired by the $k$-inversion statistic for LLT polynomials, we define a
$k$-inversion number and $k$-descent set for words. Using these, we define a
new statistic on words, called the $k$-major index, that interpolates between
the major index and inversion number. We give a bijective proof that the
$k$-major index is equidistributed with the major index, generalizing a
classical result of Foata and rediscovering a result of Kadell. Inspired by
recent work of Haglund and Stevens, we give a partial extension of these
definitions and constructions to standard Young tableaux. Finally, we give an
application to Macdonald polynomials made possible through connections with LLT
polynomials.
