Topological Narayana Polynomials and Interlacing Conjectures

Monotone Hurwitz numbers enumerate factorisations of permutations into transpositions, with a certain monotonicity constraint [3]. They arise naturally as coefficients in the large N topological expansion of the well-studied HCIZ integral over the space of N × N unitary matrices. We introduce here a deformation of the monotone Hurwitz numbers that produces a generalisation of the Narayana polynomials [1]. Numerical evidence leads to conjectures concerning real-rootedness and interlacing of these so-called topological Narayana polynomials. It is then natural to investigate whether these polynomials admit multivariate generalisations that exhibit some form of stability.