Zeros and interlacing for multiset Eulerian-Narayana polynomials

Abstract

We prove the real-rootedness conjecture of Lin, Ma, Ma, and Zhou (2021) for multiset Eulerian-Narayana polynomials, including simplicity and negativity of all zeros after removing the factor t. Using a forest decomposition and Lagrange inversion, we derive a differential formula and a first-order factorization of the differential operator. We use these factors to show that the nonconstant gamma-coefficients and ordered gamma-zeros are nondecreasing, with corresponding bounds given by the Narayana and Eulerian polynomials. The same factorization gives totally nonnegative coefficient transition matrices. We also prove strict interlacing of the zeros of generating polynomials arising from adjacent columns of the transition matrices.