  In the study of Kostka numbers and Catalan numbers, Kirillov posed a
unimodality conjecture for the rectangular Narayana polynomials. We prove that
the rectangular Narayana polynomials have only real zeros, and thereby confirm
Kirillov's unimodality conjecture with the help of Newton's inequality. By
using an equidistribution property between descent numbers and ascent numbers
on ballot paths due to Sulanke and a bijection between lattice words and
standard Young tableaux, we show that the rectangular Narayana polynomial is
equal to the descent generating function on standard Young tableaux of certain
rectangular shape, up to a power of the indeterminate. Then we obtain the
real-rootedness of the rectangular Narayana polynomial based on Brenti's result
that the descent generating function of standard Young tableaux has only real
zeros.
