Kirillov's unimodality conjecture for the rectangular Narayana polynomials

TL;DR

This paper proves that rectangular Narayana polynomials have only real zeros, confirming Kirillov's unimodality conjecture by connecting them to descent generating functions of standard Young tableaux.

Contribution

It establishes the real-rootedness of rectangular Narayana polynomials and confirms their unimodality, using combinatorial bijections and properties of standard Young tableaux.

Findings

Rectangular Narayana polynomials have only real zeros.

Confirmed Kirillov's unimodality conjecture.

Linked Narayana polynomials to descent generating functions of tableaux.

Abstract

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.