  The classical hook length formula counts the number of standard tableaux of
straight shapes. In 1996, Okounkov and Olshanski found a positive formula for
the number of standard Young tableaux of a skew shape. We prove various
properties of this formula, including three determinantal formulas for the
number of nonzero terms, an equivalence between the Okounkov-Olshanski formula
and another skew tableaux formula involving Knutson-Tao puzzles, and two
$q$-analogues for reverse plane partitions, which complements work by Stanley
and Chen for semistandard tableaux. We also give several reformulations of the
formula, including two in terms of the excited diagrams appearing in a more
recent skew tableaux formula by Naruse. Lastly, for thick zigzag shapes we show
that the number of nonzero terms is given by a determinant of the Genocchi
numbers and improve on known upper bounds by Morales-Pak-Panova on the number
of standard tableaux of these shapes.
