Top degree part in $b$-conjecture for unicellular bipartite maps


Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\psi(x,y,z; 1, 1+\beta)$ 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 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 coefficients of $\psi$ exists 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 and white, respectively, vertex degrees given by $\mu$ and $\nu$, respectively. We show that in this case this evaluation corresponds, up to a sign, to a top-degree part of the coefficients of $\psi$, and as a consequence, we are able to find a collection of integer-valued statistics of maps such that in the case of maps with only one face (called unicellular) the top-degree of their multivariate generating series 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.

Electron. J. Combin., 24 (3), Paper 3.24, 2017