Interlacing property of polynomial sequences related to multinomial coefficients

Abstract

In this paper, we show that several consecutive generating functions of the multinomial coefficients form an interlacing sequence. As applications, we provide a positive response to a question proposed by Fisk regarding the interlacing property for zeros of polynomials, which are generated by the central trinomial (quadrinomial) coefficients.

Furthermore, we prove that some classical polynomial sequences also possess the interlacing property. These sequences include the (weak) exceedance polynomial sequence on involutions in the symmetric group, Motzkin polynomial sequence, local h-polynomial sequences of the cluster subdivision of Cartan-Killing types A, B and D, Narayana polynomial sequences of types A and B, and others.