Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps

Abstract

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\psi(\boldsymbol{x}, \boldsymbol{y},\boldsymbol{z}; 1, 1+\beta)$ with an additional parameter $\beta$ that might be interpreted as a continuous deformation of the rooted bipartite maps generating series. Indeed, it has a property that for $\beta \in \{0,1\}$, it specializes to the rooted, orientable (general, i.e. orientable or not, respectively) bipartite maps generating series. They made the following conjecture: coefficients of $\psi$ are polynomials in $\beta$ with positive integer coefficients that can be written as a multivariate generating series of rooted, general bipartite maps, where the exponent of $\beta$ is an integer-valued statistics that in some sense "measures the non-orientability" of the corresponding bipartite map.We show that except two special values of $\beta = 0,1$ for which the combinatorial interpretation of the coefficients of $\psi$ is known, there exists a third special value $\beta = -1$ for which the coefficients of $\psi$ indexed by two partitions $\mu,\nu$, and one partition with only one part are given by rooted, orientable bipartite maps with arbitrary face degrees and black/white vertex degrees given by $\mu$/$\nu$, respectively. We show that this evaluation corresponds, up to a sign, to a top-degree part of the coefficients of $\psi$. As a consequence, we introduce a collection of integer-valued statistics of maps $(\eta)$ such that the top-degree of the multivariate generating series of rooted, bipartite maps with only one face (called unicellular) with respect to $\eta$ gives the top degree of the appropriate coefficients of $\psi$. Finally, we show that $b$ conjecture holds true for all rooted, unicellular bipartite maps of genus at most $2$.