Kirillov's Unimodality Conjecture for the Rectangular Narayana Polynomials
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. 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 a result of Brenti which implies that the descent generating function of standard Young tableaux has only real zeros.