Cumulants of Jack symmetric functions and bconjecture (extended abstract)
1 : Faculty of Mathematics and Computer Science [Poznan]
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznan, Poland 
Pologne
2 : Institute of Mathematics
(WROCLAW)

Website
Institute of Mathematics, University of Wroclaw, Pl. Grunwaldzki 2/4, 50384 Wroclaw, Poland 
Pologne
Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called bconjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.
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